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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"c88b5f246afde77ad2a9d71c8ac1ee06251498be2857578404b4c3fb2a3e9c44","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-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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,    1;\n  1:  1,    3,      1;\n  2:  1,    7,      6,      1;\n  3:  2,   22,     31,     11,      1;\n  4:  3,   61,    140,     85,     17,      1;\n  5:  4,  156,    571,    545,    185,     24,      1;\n  6:  6,  410,   2260,   3166,   1625,    351,     32,     1;\n  7:  9, 1079,   8758,  17325,  12621,   4039,    605,    41,    1;\n  8: 13, 2803,  33276,  90931,  90531,  40299,   8820,   972,   51,  1;\n  9: 19, 7277, 125027, 463886, 614965, 366002, 109795, 17491, 1480, 62, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-3,x) + (3*x+2*x^2)*d/dx P(n-3,x) + x^2*d^2/dx^2 P(n-3,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1 + x, P(1,x) = 1 + 3*x + x^2, P(2,x) = 1 + 7*x + 6*x^2 + x^3.","mathematica":"P[0, x_] := 1 + x;\nP[1, x_] := 1 + 3*x + x^2;\nP[2, x_] := 1 + 7*x + 6*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 3, x] + (3*x + 2*x^2)*D[P[n - 3, x], x] + x^2*D[P[n - 3, x], {x, 2}] + (1 + 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 + x; v[2] = 1 + 3*x + x^2; v[3] = 1 + 7*x + 6*x^2 + x^3; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 3] + (3*x + 2*x^2)*deriv(v[i - 3], x) + x^2*deriv(deriv(v[i - 3], x), x) + (1 + 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-0d55e0bb3c384646","steps":["reverse","basis A008277"],"edges":["e-0e9e334768ea0a35","e-535fcb4dbc81ccf5"],"text":"A102547 \u2192 reverse \u2192 basis A008277"},"headline":"A102547 \u2192 reverse \u2192 basis A008277","chain":null}