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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d0e3b69ce01408effeb872c26200dedb6a641093d10ac9cc3472ffca6430e053","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-2f02b2035f8f4541","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:           19,             8;\n  2:          361,           368,             64;\n  3:         6859,         12824,           5184,            512;\n  4:       130321,        401120,         284032,          63488,           4096;\n  5:      2476099,      11872808,       13150080,        5002240,         716800,          32768;\n  6:     47045881,     340374608,      555235264,      320296960,       76574720,        7667712,        262144;\n  7:    893871739,    9566481464,    22156231104,    18214651392,     6467686400,     1064992768,      78905344,      2097152;\n  8:  16983563041,  265445973440,   851999940352,   960479858688,   475569217536,   114576064512,   13806600192,    788529152,   16777216;\n  9: 322687697779, 7302909787208, 31943565699840, 48116633446400, 31937868963840, 10564541546496, 1841650728960, 169592487936, 7700742144, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+8*x)*P(n-1,x) + 8*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[(19 + 8*x)*P[n - 1, x] + 8*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] = (19 + 8*x)*v[i - 1] + 8*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-2f02b2035f8f4541","steps":["basis A008277"],"edges":["e-7c74a26235b2e89e"],"text":"A038322 \u2192 basis A008277"},"headline":"A038322 \u2192 basis A008277","chain":null}