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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"66597ea7799a2c5610f17d6eecd846467a7dfae341ce06e35dec1ceac9d3c6ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a3274460eb3b9452","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          1;\n  1:          2,           1;\n  2:         18,          19,           1;\n  3:         96,         195,          53,           1;\n  4:       1182,        3761,        1617,         113,           1;\n  5:      11100,       56691,       42443,        7369,         210,          1;\n  6:     187680,     1399733,     1432143,      368948,       25149,        357,        1;\n  7:    2639760,    29781207,    44668385,    17948359,     2237318,      71712,      569,      1;\n  8:   58768320,   943319991,  1831860917,   963171655,   172359772,   10517237,   180004,    863,    1;\n  9: 1133844240, 26836565283, 71328513029, 51485520277, 13214285652, 1273002581, 41099844, 409916, 1258, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/3)*n^3+(3/2)*n^2+(13/6)*n+1+((1/3)*n^3+(3/2)*n^2+(13/6)*n+1)*x)*P(n-2,x) + ((1/3)*n^3+(3/2)*n^2+(13/6)*n+1)*x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1 + ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1)*x)*P[n - 2, x] + ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1)*x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1 + ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1)*x)*v[i - 2] + ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1)*x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a3274460eb3b9452","steps":["basis A008277"],"edges":["e-239c141e75570afc"],"text":"A154228 \u2192 basis A008277"},"headline":"A154228 \u2192 basis A008277","chain":null}