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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":"b2b3a97eca6eca0b65dd7d930d702634e1d1e8d3cd529b6c47b17db23612e832","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-a13c2de84911142b","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^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,        36;\n  2:      25,       360,        729;\n  3:     125,      2700,      10935,       11664;\n  4:     625,     18000,     109350,      233280,      164025;\n  5:    3125,    112500,     911250,     2916000,     4100625,      2125764;\n  6:   15625,    675000,    6834375,    29160000,    61509375,     63772920,     26040609;\n  7:   78125,   3937500,   47840625,   255150000,   717609375,   1116026100,    911421315,    306110016;\n  8:  390625,  22500000,  318937500,  2041200000,  7176093750,  14880348000,  18228426300,  12244400640,   3486784401;\n  9: 1953125, 126562500, 2050312500, 15309000000, 64584843750, 167403915000, 273426394500, 275499014400, 156905298045, 38742048900;\n","recurrence_sentence":"The row polynomials satisfy (25+(135*n-45)*x+81*n^2*x^2)*P(n,x) = (125+(675*n+675)*x+(405*n^2+2025*n+1215)*x^2+(729*n^2+1458*n+729)*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 + (675*n + 675)*x + (405*n^2 + 2025*n + 1215)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*P[n - 1, x])/(25 + (135*n - 45)*x + 81*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 + (675*n + 675)*x + (405*n^2 + 2025*n + 1215)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*v[i - 1])/(25 + (135*n - 45)*x + 81*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-5733d79f45a14084","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-4eef4a156684ec8f","e-4c9474bfa308c0c7"],"text":"A013620 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013620 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}