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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9c6e3f8887fab697b5524046e5704aa895b458dbb2bed081673c11aaa7e67fe6","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-108b34f10b7468f4","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(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:          121,            75,              9;\n  1:         1331,          1413,            378,             27;\n  2:        14641,         23775,          10665,           1674,             81;\n  3:       161051,        376773,         252630,          65475,           6885,           243;\n  4:      1771561,       5757975,        5425029,        2067390,         354780,         26973,           729;\n  5:     19487171,      85926333,      109499418,       57622887,       14362110,       1765638,        102060,         2187;\n  6:    214358881,    1261430175,     2119269105,     1480955994,      503197191,      88992918,       8256654,       376164,        6561;\n  7:   2357947691,   18303099093,    39811865310,    35976927195,    16016403375,    3823407441,     506421720,     36807210,     1358127,     19683;\n  8:  25937424601,  263317230375,   731711007549,   838974139830,   476308059210,  147457803591,   26156452203,   2697095880,   157956075,   4822335,    59049;\n  9: 285311670611, 3764253499053, 13229038819458, 18974615819247, 13472007781320, 5262827070996, 1200910524660, 164776424769, 13619750265, 657116955, 16888014, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 121 + 75*x + 9*x^2.","mathematica":"P[0, x_] := 121 + 75*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(11 + 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] = 121 + 75*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 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-108b34f10b7468f4","steps":["basis A008277"],"edges":["e-d7f363249f6694ec"],"text":"A038281 \u2192 basis A008277"},"headline":"A038281 \u2192 basis A008277","chain":null}