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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":"c32018140a8a52ea491b92c1837901a8d790e5129298b213bfc4224153d0d9bb","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-0e9006d690eb01e9","role":"sequence","name":"basis_map_A013611 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,      48;\n   4:   29,    248,     480,      256;\n   5:   76,    920,    2880,     3328,     1280;\n   6:  199,   3120,   13680,    25088,    20480,     6144;\n   7:  521,  10032,   56880,   144896,   185600,   116736,    28672;\n   8: 1364,  31144,  217152,   712960,  1260800,  1234944,   630784,   131072;\n   9: 3571,  94312,  781440,  3152128,  7161600,  9639936,  7626752,  3276800,   589824;\n  10: 9349, 280360, 2693040, 12911104, 35984640, 62060544, 67235840, 44564480, 16515072, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(80*n-120+(128*n-192)*x-(128*n^3-576*n^2+832*n-384)*x^2)*P(n,x) = -(120*n-80+(80*n^2+272*n-288)*x-(192*n^3-704*n^2+256*n+256)*x^2-(256*n^3-768*n^2+512*n)*x^3)*P(n-1,x) - (25+200*x-(80*n^2-160*n-464)*x^2-(384*n^2-768*n-256)*x^3-(256*n^2-512*n)*x^4)*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[(-(120*n - 80 + (80*n^2 + 272*n - 288)*x - (192*n^3 - 704*n^2 + 256*n + 256)*x^2 - (256*n^3 - 768*n^2 + 512*n)*x^3)*P[n - 1, x] - (25 + 200*x - (80*n^2 - 160*n - 464)*x^2 - (384*n^2 - 768*n - 256)*x^3 - (256*n^2 - 512*n)*x^4)*D[P[n - 1, x], x])/(-80*n + 120 - (128*n - 192)*x + (128*n^3 - 576*n^2 + 832*n - 384)*x^2)]];\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] = (-(120*n - 80 + (80*n^2 + 272*n - 288)*x - (192*n^3 - 704*n^2 + 256*n + 256)*x^2 - (256*n^3 - 768*n^2 + 512*n)*x^3)*v[i - 1] - (25 + 200*x - (80*n^2 - 160*n - 464)*x^2 - (384*n^2 - 768*n - 256)*x^3 - (256*n^2 - 512*n)*x^4)*deriv(v[i - 1], x))/(-80*n + 120 - (128*n - 192)*x + (128*n^3 - 576*n^2 + 832*n - 384)*x^2)); 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 A013611","\u03b8+1"],"edges":["e-50e29512dcce7c20","e-8ca1acb44818fa13"],"text":"A111125 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A111125 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}