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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e7ffaa663f1fc760a548152d46eda6ae32d5129e3ef7b3ded5e80f7321198956","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,        1;\n   3:      5,       11,         3;\n   4:     25,      109,        72,         10;\n   5:    129,     1039,      1265,        410,         35;\n   6:    681,     9761,     19660,      11200,       2170,       126;\n   7:   3653,    91307,    287427,     256760,      84840,     10962,       462;\n   8:  19825,   854365,   4062828,    5353026,    2686110,    579810,     53592,     1716;\n   9: 108545,  8013215,  56318817,  105397194,   75427275,  24201870,   3681678,   255684,    6435;\n  10: 598417, 75396673, 771767928, 2000574400, 1963569300, 878906028, 195723528, 22119240, 1196910, 24310;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(4*n-14)*x)*P(n-2,x) + (4*n-14)*x*d/dx P(n-2,x) - (7*n-12+(4*n-6)*x)*P(n-1,x) - (4*n-6)*x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 5 + 11*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 5 + 11*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (4*n - 14)*x)*P[n - 2, x] + (4*n - 14)*x*D[P[n - 2, x], x] - (7*n - 12 + (4*n - 6)*x)*P[n - 1, x] - (4*n - 6)*x*D[P[n - 1, x], x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 5 + 11*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (4*n - 14)*x)*v[i - 2] + (4*n - 14)*x*deriv(v[i - 2], x) - (7*n - 12 + (4*n - 6)*x)*v[i - 1] - (4*n - 6)*x*deriv(v[i - 1], x))/(-n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":["basis A008277"],"edges":["e-95e572ac9568e114"],"text":"A253283 \u2192 basis A008277"},"headline":"A253283 \u2192 basis A008277","chain":null}