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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":"f064e43a614e786cf8190fb068fa8325921a116f56427d4438660e9c62c4a7b3","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-45b8d33479b2056d","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:    4,      8;\n   3:   11,     56,      36;\n   4:   29,    248,     360,     128;\n   5:   76,    920,    2160,    1664,      400;\n   6:  199,   3120,   10260,   12544,     6400,     1152;\n   7:  521,  10032,   42660,   72448,    58000,    21888,    3136;\n   8: 1364,  31144,  162864,  356480,   394000,   231552,   68992,    8192;\n   9: 3571,  94312,  586080, 1576064,  2238000,  1807488,  834176,  204800,  20736;\n  10: 9349, 280360, 2019780, 6455552, 11245200, 11636352, 7353920, 2785280, 580608, 51200;\n","recurrence_sentence":"The row polynomials satisfy (400*n-600+(320*n-480)*x-(800*n^3-3600*n^2+5264*n-2496)*x^2-(576*n^3-2592*n^2+3872*n-1920)*x^3+(64*n^5-480*n^4+1408*n^3-2016*n^2+1408*n-384)*x^4)*P(n,x) = (600*n-400+(700*n^2-20*n-520)*x-(1200*n^3-4400*n^2+3616*n-384)*x^2-(160*n^4+1264*n^3-5088*n^2+4464*n-736)*x^3+(96*n^5-640*n^4+608*n^3+1088*n^2-1408*n+256)*x^4+(64*n^5-320*n^4+512*n^3-256*n^2)*x^5)*P(n-1,x) + (125+600*x-(700*n^2-1400*n-640)*x^2-(2400*n^2-4800*n+512)*x^3+(80*n^4-320*n^3-1968*n^2+4576*n-912)*x^4+(192*n^4-768*n^3+192*n^2+1152*n-256)*x^5+(64*n^4-256*n^3+256*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((600*n - 400 + (700*n^2 - 20*n - 520)*x - (1200*n^3 - 4400*n^2 + 3616*n - 384)*x^2 - (160*n^4 + 1264*n^3 - 5088*n^2 + 4464*n - 736)*x^3 + (96*n^5 - 640*n^4 + 608*n^3 + 1088*n^2 - 1408*n + 256)*x^4 + (64*n^5 - 320*n^4 + 512*n^3 - 256*n^2)*x^5)*P[n - 1, x] + (125 + 600*x - (700*n^2 - 1400*n - 640)*x^2 - (2400*n^2 - 4800*n + 512)*x^3 + (80*n^4 - 320*n^3 - 1968*n^2 + 4576*n - 912)*x^4 + (192*n^4 - 768*n^3 + 192*n^2 + 1152*n - 256)*x^5 + (64*n^4 - 256*n^3 + 256*n^2)*x^6)*D[P[n - 1, x], x])/(400*n - 600 + (320*n - 480)*x - (800*n^3 - 3600*n^2 + 5264*n - 2496)*x^2 - (576*n^3 - 2592*n^2 + 3872*n - 1920)*x^3 + (64*n^5 - 480*n^4 + 1408*n^3 - 2016*n^2 + 1408*n - 384)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((600*n - 400 + (700*n^2 - 20*n - 520)*x - (1200*n^3 - 4400*n^2 + 3616*n - 384)*x^2 - (160*n^4 + 1264*n^3 - 5088*n^2 + 4464*n - 736)*x^3 + (96*n^5 - 640*n^4 + 608*n^3 + 1088*n^2 - 1408*n + 256)*x^4 + (64*n^5 - 320*n^4 + 512*n^3 - 256*n^2)*x^5)*v[i - 1] + (125 + 600*x - (700*n^2 - 1400*n - 640)*x^2 - (2400*n^2 - 4800*n + 512)*x^3 + (80*n^4 - 320*n^3 - 1968*n^2 + 4576*n - 912)*x^4 + (192*n^4 - 768*n^3 + 192*n^2 + 1152*n - 256)*x^5 + (64*n^4 - 256*n^3 + 256*n^2)*x^6)*deriv(v[i - 1], x))/(400*n - 600 + (320*n - 480)*x - (800*n^3 - 3600*n^2 + 5264*n - 2496)*x^2 - (576*n^3 - 2592*n^2 + 3872*n - 1920)*x^3 + (64*n^5 - 480*n^4 + 1408*n^3 - 2016*n^2 + 1408*n - 384)*x^4)); 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-318700a428d7b97b","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-0d48af5cb3d93d03","e-c1a394a78430f724"],"text":"A111125 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}