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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e7f27d06387756a9d0890add72473d309e2aaad7f43bddd5b2dd5c74b32250d2","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-b7cf68162d40a956","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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:           18,            44;\n  2:          324,          1584,           1936;\n  3:         5832,         42768,         104544,           85184;\n  4:       104976,       1026432,        3763584,         6133248,         3748096;\n  5:      1889568,      23094720,      112907520,       275996160,       337328640,        164916224;\n  6:     34012224,     498845952,     3048503040,      9935861760,     18215746560,      17810952192,       7256313856;\n  7:    612220032,   10475764992,    76822276608,    312979645440,    765061355520,    1122089988096,     914295545856,     319277809664;\n  8:  11019960576,  215501451264,  1843734638592,   9013813788672,  27542208798720,   53860319428608,   65829279301632,   45976004591616,   14048223625216;\n  9: 198359290368, 4363904388096, 42669287350272, 243372972294144, 892367565078528, 2181342936858624, 3554781082288128, 3724056371920896, 2275812227284992, 618121839509504;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(18 + 44*x)*P[n - 1, 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] = (18 + 44*x)*v[i - 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-b7cf68162d40a956","steps":["basis A013611"],"edges":["e-f0378e56b9f5156d"],"text":"A038277 \u2192 basis A013611"},"headline":"A038277 \u2192 basis A013611","chain":null}