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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"2d89b1e4e255a59d94c747f1cc993c76fb3c0ed38f3c5d002903254b786ffe23","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-f364085508d20047","role":"sequence","name":"Triangle T(n,k) read by rows; related to number of preorders.","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,          2;\n  2:        9,         21,           6;\n  3:       44,        228,         164,           24;\n  4:      265,       2755,        3795,         1330,          120;\n  5:     1854,      37650,       89160,        56250,        11724,          720;\n  6:    14833,     580503,     2234890,      2262925,       817600,       113148,        5040;\n  7:   133496,   10023944,    60722424,     92806000,     51726360,     12100256,     1193184,      40320;\n  8:  1334961,  192215007,  1795287501,   3989097126,   3220098525,   1132419456,   185408748,   13687056,    362880;\n  9: 14684570, 4059463590, 57702687260, 181836224290, 204410275510, 101389431570, 24507332360, 2964209400, 169932960, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (n+1)*x*d/dx P(n-2,x) + (n+1+(n+1)*x)*P(n-1,x) + (n+1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1)*x*D[P[n - 2, x], x] + (n + 1 + (n + 1)*x)*P[n - 1, x] + (n + 1)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1)*x*deriv(v[i - 2], x) + (n + 1 + (n + 1)*x)*v[i - 1] + (n + 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-f364085508d20047","steps":["basis A008277"],"edges":["e-9bc96d65e7435b07"],"text":"A079510 \u2192 basis A008277"},"headline":"A079510 \u2192 basis A008277","chain":null}