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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"628e58f8ee9e30d8a3412012c280c8f8e4f7a9d13482407816e0179dff142c5c","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-81e6fb4f9f226d77","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          13,           1;\n  2:         169,          27,          1;\n  3:        2197,         547,         42,         1;\n  4:       28561,        9855,       1177,        58,        1;\n  5:      371293,      166531,      27510,      2105,       75,       1;\n  6:     4826809,     2702727,     579181,     61190,     3380,      93,      1;\n  7:    62748517,    42664987,   11390442,   1558221,   118650,    5054,    112,    1;\n  8:   815730721,   660058335,  213521617,  36321978,  3575271,  209622,   7182,  132,   1;\n  9: 10604499373, 10056547411, 3862882590, 794673265, 97101585, 7348467, 346080, 9822, 153, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(13 + 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; for(n = 1, N, my(i = n + 1); v[i] = (13 + 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-81e6fb4f9f226d77","steps":["basis A008277"],"edges":["e-04b842e58817f21f"],"text":"A038327 \u2192 basis A008277"},"headline":"A038327 \u2192 basis A008277","chain":null}