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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9a2f964bfd34d7160fba97eb7dbccd848b8758b65c496f6d68a8217eae0032fe","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-2ed1083d14adf57b","role":"sequence","name":"Mirror of the triangle A193726.","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:        3,         1;\n  2:       21,        15,         2;\n  3:      147,       177,        52,         4;\n  4:     1029,      1887,       926,       156,         8;\n  5:     7203,     19041,     13960,      3880,       432,       16;\n  6:    50421,    185775,    191642,     78360,     14240,     1136,      32;\n  7:   352947,   1772817,   2479612,   1401964,    370320,    47792,    2880,     64;\n  8:  2470629,  16661247,  30821366,  23184756,   8358728,  1553104,  150304,   7104,   128;\n  9: 17294403, 154892481, 372357520, 363044560, 171750432, 43120224, 5961984, 449792, 17152, 256;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(2*n-2)*x)*P(n-1,x) + (2*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (2*n - 2)*x)*P[n - 1, x] + (2*n - 2)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (2*n - 2)*x)*v[i - 1] + (2*n - 2)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2ed1083d14adf57b","steps":["basis A008277"],"edges":["e-edfde46cfa2a5452"],"text":"A193727 \u2192 basis A008277"},"headline":"A193727 \u2192 basis A008277","chain":null}