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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"14604bdfcbfd9074d12484550255ed66a8e9317b90724c07eeca5f7bb705e44a","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-4397e742863fdbe2","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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:           169,             87,              9;\n  1:          2197,           1899,            432,             27;\n  2:         28561,          36975,          13905,           1890,             81;\n  3:        371293,         677283,         375120,          83295,           7695,           243;\n  4:       4826809,       11950407,        9159129,        2957850,         442260,         29889,           729;\n  5:      62748517,      205686939,      209874672,       92550087,       19930050,       2163672,        112266,         2187;\n  6:     815730721,     3479236575,     4604679585,     2665725930,      775901511,     120372966,       9971262,       411156,        6561;\n  7:   10604499373,    58114977363,    97926621840,    72460009215,    27394715565,    5698147581,     670228020,     43893090,     1476225,     19683;\n  8:  137858491849,   961653135927,  2034950747049,  1887900068250,   902247916770,  241732278963,   37871511363,   3503049120,   186299595,   5215995,    59049;\n  9: 1792160394037, 15800025650379, 41549023601712, 47638653742647, 28219898124000, 9475247561274, 1899213689142, 232718204169, 17402232375, 767538585, 18187092, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 169 + 87*x + 9*x^2.","mathematica":"P[0, x_] := 169 + 87*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(13 + 3*x)*P[n - 1, x] + 3*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] = 169 + 87*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 3*x)*v[i - 1] + 3*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-4397e742863fdbe2","steps":["basis A008277"],"edges":["e-1eab37781c60e76a"],"text":"A038305 \u2192 basis A008277"},"headline":"A038305 \u2192 basis A008277","chain":null}