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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":"7ea996c77bcfd1f99009e3035ed027f7f3d75cd36a9e7f7662bc20d7f06e5115","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-b7cf68162d40a956","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","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:        7,         44;\n  2:       49,        616,        1089;\n  3:      343,       6468,       22869,        21296;\n  4:     2401,      60368,      320166,       596288,       366025;\n  5:    16807,     528220,     3735270,     10435040,     12810875,       5797836;\n  6:   117649,    4437048,    39220335,    146090560,    269028375,     243509112,      86806489;\n  7:   823543,   36235892,   384359283,   1789609360,   4394130125,    5965973244,    4253517961,    1247178944;\n  8:  5764801,  289887136,  3587353308,  20043624832,  61517821750,  111364833888,  119098502908,   69842020864,   17363069361;\n  9: 40353607, 2282861196, 32286179772, 210458060736, 775124554050, 1753996133736, 2501068561068, 2200023657216, 1093873369743, 235794769100;\n","recurrence_sentence":"The row polynomials satisfy (49+(231*n-77)*x+121*n^2*x^2)*P(n,x) = (343+(1617*n+1617)*x+(847*n^2+4235*n+2541)*x^2+(1331*n^2+2662*n+1331)*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[((343 + (1617*n + 1617)*x + (847*n^2 + 4235*n + 2541)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*P[n - 1, x])/(49 + (231*n - 77)*x + 121*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] = ((343 + (1617*n + 1617)*x + (847*n^2 + 4235*n + 2541)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*v[i - 1])/(49 + (231*n - 77)*x + 121*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-b7cf68162d40a956","steps":["(\u03b8+1)\u00b2"],"edges":["e-a104487c663ed748"],"text":"A038277 \u2192 (\u03b8+1)\u00b2"},"headline":"A038277 \u2192 (\u03b8+1)\u00b2","chain":null}