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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"c32a5b78d3da3051a69b23f191aa762f5692470c6a88e2737b8d39d84bf399be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-210f23d82ddfabd8","role":"sequence","name":"basis_map_A013609 of 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,        32;\n  2:      25,       320,        576;\n  3:     125,      2400,       8640,        8192;\n  4:     625,     16000,      86400,      163840,      102400;\n  5:    3125,    100000,     720000,     2048000,     2560000,     1179648;\n  6:   15625,    600000,    5400000,    20480000,    38400000,    35389440,     12845056;\n  7:   78125,   3500000,   37800000,   179200000,   448000000,   619315200,    449576960,    134217728;\n  8:  390625,  20000000,  252000000,  1433600000,  4480000000,  8257536000,   8991539200,   5368709120,  1358954496;\n  9: 1953125, 112500000, 1620000000, 10752000000, 40320000000, 92897280000, 134873088000, 120795955200, 61152952320, 13421772800;\n","recurrence_sentence":"The row polynomials satisfy (25+(120*n-40)*x+64*n^2*x^2)*P(n,x) = (125+(600*n+600)*x+(320*n^2+1600*n+960)*x^2+(512*n^2+1024*n+512)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((125 + (600*n + 600)*x + (320*n^2 + 1600*n + 960)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(25 + (120*n - 40)*x + 64*n^2*x^2)]];\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] = ((125 + (600*n + 600)*x + (320*n^2 + 1600*n + 960)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(25 + (120*n - 40)*x + 64*n^2*x^2)); 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-01cea33b6485d9cc","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-391376c704e767eb","e-675350e5b07681ef"],"text":"A038220 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038220 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}