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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":"874fbd20b774855ceb50d743786e2b04cb4b24117339395ad9ae0f256dec6f15","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-f3a4ae052af3da90","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.","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,         20;\n  2:      25,        320,          225;\n  3:     125,       3660,         8235,          2000;\n  4:     625,      36280,       181170,        145120,         15625;\n  5:    3125,     332740,      3135690,       5574560,       2079625,        112500;\n  6:   15625,    2911920,     47219895,     156087680,     131166375,      26207280,        765625;\n  7:   78125,   24732860,    650455245,    3612317680,    5644246375,    2601820980,     302977535,      5000000;\n  8:  390625,  205987880,   8434933200,   73699745120,  191904376750,  165824426520,   45923525200,   3295806080,    31640625;\n  9: 1953125, 1693215540, 104834556180, 1376691478080, 5565491439750, 8014307673240, 4216117651620, 745490177280, 34287614685, 195312500;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+12*n+4)*P(n,x) = -(90*n^2+210*n+100+(540*n^3+1080*n^2+396*n-272)*x-(729*n^4+1782*n^3+1053*n^2-108*n-156)*x^2)*P(n-2,x) - ((270*n^2+630*n+444)*x-(540*n^3+1944*n^2+2277*n+1084)*x^2-(324*n^4-270*n^3-2457*n^2-2532*n-740)*x^3)*d/dx P(n-2,x) - ((81*n^2+189*n+144)*x^2-(324*n^3+972*n^2+945*n+348)*x^3+(324*n^3+891*n^2+756*n+204)*x^4)*d^2/dx^2 P(n-2,x) + (63*n^2+102*n+40+(108*n^3+216*n^2+144*n+32)*x)*P(n-1,x) + ((54*n^2+99*n+60)*x-(108*n^3+324*n^2+288*n+80)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 20*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(90*n^2 + 210*n + 100 + (540*n^3 + 1080*n^2 + 396*n - 272)*x - (729*n^4 + 1782*n^3 + 1053*n^2 - 108*n - 156)*x^2)*P[n - 2, x] - ((270*n^2 + 630*n + 444)*x - (540*n^3 + 1944*n^2 + 2277*n + 1084)*x^2 - (324*n^4 - 270*n^3 - 2457*n^2 - 2532*n - 740)*x^3)*D[P[n - 2, x], x] - ((81*n^2 + 189*n + 144)*x^2 - (324*n^3 + 972*n^2 + 945*n + 348)*x^3 + (324*n^3 + 891*n^2 + 756*n + 204)*x^4)*D[P[n - 2, x], {x, 2}] + (63*n^2 + 102*n + 40 + (108*n^3 + 216*n^2 + 144*n + 32)*x)*P[n - 1, x] + ((54*n^2 + 99*n + 60)*x - (108*n^3 + 324*n^2 + 288*n + 80)*x^2)*D[P[n - 1, x], x])/(9*n^2 + 12*n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(90*n^2 + 210*n + 100 + (540*n^3 + 1080*n^2 + 396*n - 272)*x - (729*n^4 + 1782*n^3 + 1053*n^2 - 108*n - 156)*x^2)*v[i - 2] - ((270*n^2 + 630*n + 444)*x - (540*n^3 + 1944*n^2 + 2277*n + 1084)*x^2 - (324*n^4 - 270*n^3 - 2457*n^2 - 2532*n - 740)*x^3)*deriv(v[i - 2], x) - ((81*n^2 + 189*n + 144)*x^2 - (324*n^3 + 972*n^2 + 945*n + 348)*x^3 + (324*n^3 + 891*n^2 + 756*n + 204)*x^4)*deriv(deriv(v[i - 2], x), x) + (63*n^2 + 102*n + 40 + (108*n^3 + 216*n^2 + 144*n + 32)*x)*v[i - 1] + ((54*n^2 + 99*n + 60)*x - (108*n^3 + 324*n^2 + 288*n + 80)*x^2)*deriv(v[i - 1], x))/(9*n^2 + 12*n + 4)); 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-f3a4ae052af3da90","steps":["(\u03b8+1)\u00b2"],"edges":["e-cffab39fb4c91f0a"],"text":"A257624 \u2192 (\u03b8+1)\u00b2"},"headline":"A257624 \u2192 (\u03b8+1)\u00b2","chain":null}