[1] Lemma 2.1(b): 300 random pairs (p, f), deg p <= 6, deg f <= 30, all n: 0 mismatches; the wrong formula with C(n,i) differs in 2480 cases (control)
[2] (3) for 1 <= r <= 40, r-1 <= u < 200; N! >= (N/e)^N for N <= 400; (j-1)^N/N! <= 2^-N for j <= 100, ceil(2e(j-1)) <= N < ceil(2e(j-1)) + 60: PASS
[3] C_s contained in C_(s+1) for s <= 8 (so every member of P recurs in the size enumeration); all members used lie in P: PASS
[4,6] size: J = 120; degrees of g_1..g_30: [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
      Construction A: (A1), (A2), disjoint blocks: PASS; n_1..n_12 = [2, 5, 12, 21, 34, 50, 69, 92, 119, 149, 182, 219]; n_J = 23770
      Construction B: (B1), (B2): PASS; (L1), (L2): PASS; n_1..n_12 = [3, 11, 25, 45, 72, 106, 146, 193, 247, 308, 376, 451]; n_J = 54162
      explicit choices after (A2) and (B2) satisfy (A2) and (B2) for all j <= J: PASS
[5] size: Theorem A: (z f_A)^(n)(0) = n a_(n-1) in {0} u {|w| >= 2} for 0 <= n <= 23772 (192 non-zero values): PASS; monomials c z^d for (c, d) = (1,2), (1/2,1), (3i,3), ((1-i)/7,5), ((3+4i)/5000,1), ((5-12i)/13,12): values in {0} u {|w| >= 2|c|}: PASS; Remark 3.3 (n a_n): PASS
[7] size: p = z: d = 1, S = 1, c_min = 1, j_0 = 6, n_* = 100; n_* <= n <= 54164: 7432 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 49); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 1 + z: d = 1, S = 2, c_min = 1, j_0 = 8, n_* = 185; n_* <= n <= 54164: 7530 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 68.6); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 1/5 + z: d = 1, S = 6/5, c_min = 1/5, j_0 = 32, n_* = 3461; n_* <= n <= 54164: 6991 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.16e+03); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] size: p = 1/3 + 2z: d = 1, S = 7/3, c_min = 1/3, j_0 = 26, n_* = 2241; n_* <= n <= 54164: 7180 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.05e+03); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] size: p = 1 - z + z^2: d = 2, S = 3, c_min = 1, j_0 = 10, n_* = 298; n_* <= n <= 54165: 7619 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.93e+04); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 2 + z^3: d = 3, S = 3, c_min = 1, j_0 = 10, n_* = 298; n_* <= n <= 54166: 7730 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 6.02e+06); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = z + z^2 + z^3: d = 3, S = 3, c_min = 1, j_0 = 10, n_* = 298; n_* <= n <= 54166: 7619 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 6.02e+06); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = (3+4i)/5 + z^2: d = 2, S = 2, c_min = 1, j_0 = 8, n_* = 185; n_* <= n <= 54165: 7643 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.34e+04); below n_*: 1 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 1 + (i/2) z + z^5/4: d = 5, S = 7/4, c_min = 1/4, j_0 = 30, n_* = 3024; n_* <= n <= 54168: 7422 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 2.51e+16); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 1/3 + (8+15i)/17 z: d = 1, S = 4/3, c_min = 1/3, j_0 = 20, n_* = 1293; n_* <= n <= 54164: 7333 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 394); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] size: p = 1 + (5-12i)/13 z^4: d = 4, S = 2, c_min = 1, j_0 = 8, n_* = 185; n_* <= n <= 54167: 7869 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 5.24e+05); below n_*: 3 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 100 z: d = 1, S = 100, c_min = 100, j_0 = 3, n_* = 22; n_* <= n <= 54164: 7447 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 36); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] size: p = 7 - 5z + z^3/2: d = 3, S = 25/2, c_min = 1/2, j_0 = 58, n_* = 11952; n_* <= n <= 54166: 5882 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 5.98e+10); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[8] size: f_A, p = 1 + z: a_(n_j - 1) = 0 for 120 of the 120 blocks, and there s_(n_j) = g_j(0): PASS; values n <= 23771 with s_n not in {0} u {|w| >= 2}: 96, of which 96 at block starts and 15 in the disc |s_n - 1| < 1
[9] size: sum_(i=j+1..J) T_i <= 2^-j for 1 <= j < J: PASS (largest ratio to 2^-j: A 0.807, B 0.821)
[10] size: exact majorants of the zone-i (block-i) part of f^(n_j) on |z| <= j, 1 <= j <= 5, j < i <= j + 6: all <= 2^-i: PASS (largest ratio 9.335e-01)
[4,6] mix: J = 120; degrees of g_1..g_30: [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 1, 0, 2, 3, 4, 1, 2, 0, 0, 0, 0, 0, 0]
      Construction A: (A1), (A2), disjoint blocks: PASS; n_1..n_12 = [14, 17, 23, 33, 46, 63, 83, 106, 134, 164, 198, 236]; n_J = 23797
      Construction B: (B1), (B2): PASS; (L1), (L2): PASS; n_1..n_12 = [8, 16, 30, 50, 77, 138, 178, 225, 280, 341, 409, 485]; n_J = 65481
      explicit choices after (A2) and (B2) satisfy (A2) and (B2) for all j <= J: PASS
[5] mix: Theorem A: (z f_A)^(n)(0) = n a_(n-1) in {0} u {|w| >= 2} for 0 <= n <= 23798 (242 non-zero values): PASS; monomials c z^d for (c, d) = (1,2), (1/2,1), (3i,3), ((1-i)/7,5), ((3+4i)/5000,1), ((5-12i)/13,12): values in {0} u {|w| >= 2|c|}: PASS; Remark 3.3 (n a_n): PASS
[7] mix: p = z: d = 1, S = 1, c_min = 1, j_0 = 6, n_* = 132; n_* <= n <= 65482: 7480 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 4.24); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1 + z: d = 1, S = 2, c_min = 1, j_0 = 8, n_* = 217; n_* <= n <= 65482: 7575 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 8.77); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1/5 + z: d = 1, S = 6/5, c_min = 1/5, j_0 = 32, n_* = 14781; n_* <= n <= 65482: 7006 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 4.78e+03); below n_*: 2 values not in {0} u {|w| >= 2}, 2 in the disc |w - 1| < 1
[7] mix: p = 1/3 + 2z: d = 1, S = 7/3, c_min = 1/3, j_0 = 26, n_* = 13563; n_* <= n <= 65482: 7189 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 6.27e+03); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1 - z + z^2: d = 2, S = 3, c_min = 1, j_0 = 10, n_* = 331; n_* <= n <= 65483: 7661 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 3.36e+03); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] mix: p = 2 + z^3: d = 3, S = 3, c_min = 1, j_0 = 10, n_* = 331; n_* <= n <= 65484: 7772 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.15e+06); below n_*: 2 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = z + z^2 + z^3: d = 3, S = 3, c_min = 1, j_0 = 10, n_* = 331; n_* <= n <= 65484: 7661 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 1.15e+06); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] mix: p = (3+4i)/5 + z^2: d = 2, S = 2, c_min = 1, j_0 = 8, n_* = 217; n_* <= n <= 65483: 7688 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 2.01e+03); below n_*: 1 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1 + (i/2) z + z^5/4: d = 5, S = 7/4, c_min = 1/4, j_0 = 30, n_* = 14345; n_* <= n <= 65486: 7435 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 4.41e+19); below n_*: 2 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1/3 + (8+15i)/17 z: d = 1, S = 4/3, c_min = 1/3, j_0 = 20, n_* = 12614; n_* <= n <= 65482: 7348 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 497); below n_*: 2 values not in {0} u {|w| >= 2}, 1 in the disc |w - 1| < 1
[7] mix: p = 1 + (5-12i)/13 z^4: d = 4, S = 2, c_min = 1, j_0 = 8, n_* = 217; n_* <= n <= 65485: 7914 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 2.62e+05); below n_*: 3 values not in {0} u {|w| >= 2}, 2 in the disc |w - 1| < 1
[7] mix: p = 100 z: d = 1, S = 100, c_min = 100, j_0 = 3, n_* = 27; n_* <= n <= 65482: 7497 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 7.14); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[7] mix: p = 7 - 5z + z^3/2: d = 3, S = 25/2, c_min = 1/2, j_0 = 58, n_* = 23274; n_* <= n <= 65484: 5898 non-zero values, all with |s_n| >= S + 2: PASS (min |s_n|/(S+2) = 4.38e+11); below n_*: 0 values not in {0} u {|w| >= 2}, 0 in the disc |w - 1| < 1
[8] mix: f_A, p = 1 + z: a_(n_j - 1) = 0 for 120 of the 120 blocks, and there s_(n_j) = g_j(0): PASS; values n <= 23797 with s_n not in {0} u {|w| >= 2}: 110, of which 110 at block starts and 29 in the disc |s_n - 1| < 1
[9] mix: sum_(i=j+1..J) T_i <= 2^-j for 1 <= j < J: PASS (largest ratio to 2^-j: A 0.875, B 0.817)
[10] mix: exact majorants of the zone-i (block-i) part of f^(n_j) on |z| <= j, 1 <= j <= 5, j < i <= j + 6: all <= 2^-i: PASS (largest ratio 4.741e-01)
checks: 644, failed: 0; elapsed 13.7 s
ALL CHECKS PASSED
