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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9d8f4f485581ece40482e09dd96c8ff8247c103ad32efe7a9f4c58bf6a0849ad","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-3272d5bc697b9cb4","role":"sequence","name":"Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= 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:      2,       1;\n   3:      5,       6,        1;\n   4:     16,      33,       12,       1;\n   5:     65,     196,      114,      20,       1;\n   6:    326,    1305,     1060,     290,      30,      1;\n   7:   1957,    9786,    10275,    3860,     615,     42,     1;\n   8:  13700,   82201,   106176,   50995,   11060,   1155,    56,    1;\n   9: 109601,  767208,  1178212,  691096,  190470,  26936,  1988,   72,  1;\n  10: 986410, 7891281, 14056344, 9754500, 3269196, 585270, 58296, 3204, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["basis A008277"],"edges":["e-58b88bdc0a8c36fd"],"text":"A094816 \u2192 basis A008277"},"headline":"A094816 \u2192 basis A008277","chain":null}