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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":"fd0cf6213a1877c4aa8e21ac4d39025c4b3881779741ed3b0582f85ba4b21683","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-31f0871ac8eccdd6","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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:         11,          4;\n  2:        121,         88,          9;\n  3:       1331,       1452,        297,         16;\n  4:      14641,      21296,       6534,        704,        25;\n  5:     161051,     292820,     119790,      19360,      1375,       36;\n  6:    1771561,    3865224,    1976535,     425920,     45375,     2376,      49;\n  7:   19487171,   49603708,   30438639,    8198960,   1164625,    91476,    3773,     64;\n  8:  214358881,  623589472,  446433372,  144301696,  25621750,  2683296,  166012,   5632,   81;\n  9: 2357947691, 7716919716, 6313843404, 2380977984, 507310650, 66411576, 5478396, 278784, 8019, 100;\n","recurrence_sentence":"The row polynomials satisfy (121+(33*n-11)*x+n^2*x^2)*P(n,x) = (1331+(363*n+363)*x+(11*n^2+55*n+33)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1331 + (363*n + 363)*x + (11*n^2 + 55*n + 33)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(121 + (33*n - 11)*x + n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (363*n + 363)*x + (11*n^2 + 55*n + 33)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(121 + (33*n - 11)*x + n^2*x^2)); 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-a920b56c9b3ef4d1","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-34d290d6207a0155","e-8ec2000fabf8c353"],"text":"A038303 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038303 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}