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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":"baeab708249a4d45854645e74d5c046bc986ddc8c689500507dbfbc5956896cc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         1,           8;\n   3:         4,          76,            36;\n   4:        28,         888,          1323,           128;\n   5:       280,       12776,         37152,         14192,           400;\n   6:      3640,      220096,       1022094,        840608,        120875,          1152;\n   7:     58240,     4429344,      29419092,      41002816,      13892100,        906012,         3136;\n   8:   1106560,   102104768,     904899708,    1880287360,    1135391000,     190841400,      6281947,        8192;\n   9:  24344320,  2654422720,   29931906384,   85636988224,   79734580000,   25045725240,   2329328680,    41377216,     20736;\n  10: 608608000, 76855645440, 1065255473664, 3975903159936, 5224911959700, 2609278142400, 475151297250, 26242459776, 262856259, 51200;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-24*n+16)*P(n,x) = -(81*n^4-648*n^3+1683*n^2-1548*n+352+(324*n^4-2160*n^3+5184*n^2-6024*n+3668)*x-(729*n^4-4050*n^3+7857*n^2-6264*n+1680)*x^2)*P(n-2,x) - ((162*n^3-621*n^2+360*n+303)*x-(324*n^4-1836*n^3+3780*n^2-2550*n-937)*x^2-(324*n^4-2862*n^3+6939*n^2-6312*n+1840)*x^3)*d/dx P(n-2,x) - ((81*n^2-54*n-261)*x^2-(324*n^3-972*n^2+1026*n-597)*x^3+(324*n^3-1053*n^2+1080*n-336)*x^4)*d^2/dx^2 P(n-2,x) + (54*n^3-261*n^2+345*n-124+(108*n^3-432*n^2+576*n-256)*x)*P(n-1,x) + ((54*n^2-90*n-39)*x-(108*n^3-324*n^2+288*n-64)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*P[n - 2, x] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*D[P[n - 2, x], {x, 2}] + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*P[n - 1, x] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 24*n + 16)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = (-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*v[i - 2] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*deriv(deriv(v[i - 2], x), x) + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*v[i - 1] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 24*n + 16)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d0e888e78d074358","steps":["(\u03b8+1)\u00b2"],"edges":["e-84c08d15a4521260"],"text":"A290595 \u2192 (\u03b8+1)\u00b2"},"headline":"A290595 \u2192 (\u03b8+1)\u00b2","chain":null}