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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"487b15cb48a059dc844d0a6c0258ae88d52ee0cd917306323634f71b27b77110","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-6a5af5e1168bb679","role":"sequence","name":"Triangle of coefficients in expansion of (1+3*x)^n.","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:      4,        3;\n  2:     16,       33,         9;\n  3:     64,      279,       189,         27;\n  4:    256,     2145,      2727,        918,         81;\n  5:   1024,    15783,     33705,      20115,       4050,       243;\n  6:   4096,   113553,    384399,     362610,     125145,     16767,       729;\n  7:  16384,   807159,   4184649,    5867127,    3090150,    694008,     66339,     2187;\n  8:  65536,  5699265,  44267967,   88826598,   67043781,  22456602,   3541482,   253692,   6561;\n  9: 262144, 40091463, 459777465, 1287549675, 1339180290, 627806781, 145282410, 16966746, 944784, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + 3*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[(4 + 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; for(n = 1, N, my(i = n + 1); v[i] = (4 + 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-6a5af5e1168bb679","steps":["basis A008277"],"edges":["e-240eb6cab1dce1b5"],"text":"A013610 \u2192 basis A008277"},"headline":"A013610 \u2192 basis A008277","chain":null}