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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":"00f65b13c972519ec7d843d7cce0331883cacf5951a852c688a89da2e9773d7c","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-7b068e1328e63a95","role":"sequence","name":"basis_map_A013610 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,       24;\n  2:      25,      240,        432;\n  3:     125,     1800,       6480,       6912;\n  4:     625,    12000,      64800,     138240,      103680;\n  5:    3125,    75000,     540000,    1728000,     2592000,      1492992;\n  6:   15625,   450000,    4050000,   17280000,    38880000,     44789760,     20901888;\n  7:   78125,  2625000,   28350000,  151200000,   453600000,    783820800,    731566080,    286654464;\n  8:  390625, 15000000,  189000000, 1209600000,  4536000000,  10450944000,  14631321600,  11466178560,   3869835264;\n  9: 1953125, 84375000, 1215000000, 9072000000, 40824000000, 117573120000, 219469824000, 257989017600, 174142586880, 51597803520;\n","recurrence_sentence":"The row polynomials satisfy (5+12*n*x)*P(n,x) = (25+(60*n+120)*x+(144*n+144)*x^2)*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[((25 + (60*n + 120)*x + (144*n + 144)*x^2)*P[n - 1, x])/(5 + 12*n*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] = ((25 + (60*n + 120)*x + (144*n + 144)*x^2)*v[i - 1])/(5 + 12*n*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-5733d79f45a14084","steps":["basis A013611","\u03b8+1"],"edges":["e-1a3f71d8de8c31ec","e-45d818a2c4fb907d"],"text":"A013620 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A013620 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}