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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e9f25ac5927ca614735cde132a29e3f14429f6550c41aab707af2a48fdb6c69f","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-d07b286c67e2d8fc","role":"sequence","name":"basis_map_A013610 of Pascal-(1,4,1) array.","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:        5,          3;\n  2:       41,         51,          9;\n  3:      285,        675,        288,         27;\n  4:     2081,       8175,       6165,       1350,         81;\n  5:    14965,      93963,     111960,      42255,       5751,        243;\n  6:   108121,    1050471,    1858149,    1071630,     245916,      23085,       729;\n  7:   780045,   11538675,   29197008,   24118047,    8270586,    1286928,     88938,     2187;\n  8:  5630161,  125360175,  442714725,  503266230,  241552287,   55255770,   6247530,   332424,    6561;\n  9: 40631525, 1352215323, 6552286920, 9978373935, 6434921637, 2026825821, 332730180, 28671570, 1213785, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+12*x)*P(n-2,x) + 12*x*d/dx P(n-2,x) + (5+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(16 + 12*x)*P[n - 2, x] + 12*x*D[P[n - 2, x], x] + (5 + 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] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (16 + 12*x)*v[i - 2] + 12*x*deriv(v[i - 2], x) + (5 + 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-ed9434bfd889c409","steps":["basis A013610","basis A008277"],"edges":["e-14634eb9644b375c","e-1bfffcb068d8cc10"],"text":"A081579 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A081579 \u2192 basis A013610 \u2192 basis A008277","chain":null}