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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"1faa52b11c4a893ae4d5e89a481d9869f8085073bfd769e1dbafdb8d67004f81","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-00bc3b0d8b47bf75","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:         2,          12;\n  2:         8,         120,          243;\n  3:        48,        1200,         5508,          6480;\n  4:       384,       13632,       100764,        257472,         212625;\n  5:      3840,      177792,      1835784,       7411392,       12988350,        8266860;\n  6:     46080,     2640384,     35230464,     196636032,      538010100,      716408712,       371319795;\n  7:    645120,    44153856,    724920192,    5195549952,    19514649600,    40244369184,     43227839592,     18914575680;\n  8:  10321920,   822583296,  16069300992,  141122414592,   677546499600,  1908459201024,   3156613045992,   2843371537920,   1077244193025;\n  9: 185794560, 16912171008, 383638851072, 3997153419264, 23441529405600, 84111280549056, 189297911032272, 261449768950272, 202921142711130, 67826486227500;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(96*n^3-288*n^2+264*n-72)*x-(648*n^3-2268*n^2+2430*n-729)*x^2)*P(n-2,x) + ((96*n^3-288*n^2+264*n-72)*x^2+(288*n^3-1008*n^2+1080*n-324)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+2+(48*n^2-96*n+36)*x)*P(n-1,x) - (48*n^2-96*n+36)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (96*n^3 - 288*n^2 + 264*n - 72)*x - (648*n^3 - 2268*n^2 + 2430*n - 729)*x^2)*P[n - 2, x] + ((96*n^3 - 288*n^2 + 264*n - 72)*x^2 + (288*n^3 - 1008*n^2 + 1080*n - 324)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 2 + (48*n^2 - 96*n + 36)*x)*P[n - 1, x] - (48*n^2 - 96*n + 36)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (96*n^3 - 288*n^2 + 264*n - 72)*x - (648*n^3 - 2268*n^2 + 2430*n - 729)*x^2)*v[i - 2] + ((96*n^3 - 288*n^2 + 264*n - 72)*x^2 + (288*n^3 - 1008*n^2 + 1080*n - 324)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 2 + (48*n^2 - 96*n + 36)*x)*v[i - 1] - (48*n^2 - 96*n + 36)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-7e513916b57d5183","e-745c686cc4e75ea2"],"text":"A109692 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A109692 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}