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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"de9b4256d1782ed4fe6793a04272c47e5c654a43ae3325ca0d17ef08177f0734","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-2e0409c0014ec240","role":"sequence","name":"Triangle of coefficients in expansion of (1+4x)^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:       5,         4;\n  2:      25,        56,         16;\n  3:     125,       604,        432,          64;\n  4:     625,      5936,       8032,        2816,         256;\n  5:    3125,     55924,     128160,       80000,       16640,       1024;\n  6:   15625,    515816,    1889776,     1872640,      669440,      92160,       4096;\n  7:   78125,   4704844,   26630352,    39393984,    21548800,    4981760,     487424,     16384;\n  8:  390625,  42656096,  365013952,   776219136,   610100736,  210739200,   34062336,   2490368,    65536;\n  9: 1953125, 385467364, 4915805760, 14655781120, 15916992000, 7708882944, 1830764544, 218431488, 12386304, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) + 4*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[(5 + 4*x)*P[n - 1, x] + 4*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] = (5 + 4*x)*v[i - 1] + 4*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-2e0409c0014ec240","steps":["basis A008277"],"edges":["e-dd742906c5111887"],"text":"A013611 \u2192 basis A008277"},"headline":"A013611 \u2192 basis A008277","chain":null}