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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":"1db0f1a9ab6ed50ccf8b7c93c3a4fc04604bca6b9db5cabf8af443a0e996f157","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-f7cf3d554e6e9f71","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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,        8;\n  2:      24,       80,        48;\n  3:     115,      584,       720,       256;\n  4:     551,     3760,      7056,      5120,      1280;\n  5:    2640,    22624,     57120,     62976,     32000,      6144;\n  6:   12649,   130480,    414288,    614400,    473600,    184320,     28672;\n  7:   60605,   730968,   2797200,   5218560,   5408000,   3188736,   1003520,    131072;\n  8:  290376,  4009200,  17957520,  40396800,  52659200,  41717760,  19869696,   5242880,   589824;\n  9: 1391275, 21638040, 111045600, 292538880, 459872000, 458164224, 293027840, 116916224, 26542080, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(336*n+160*n*x-128*n^3*x^2)*P(n,x) = -(840*n+840+(336*n^2+1744*n+1408)*x-(320*n^3+320*n^2-640*n-640)*x^2-(256*n^3+512*n^2+256*n)*x^3)*P(n-1,x) - (441+1260*x-(336*n^2+336*n-1136)*x^2-(640*n^2+640*n-320)*x^3-(256*n^2+256*n)*x^4)*d/dx 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[(-(840*n + 840 + (336*n^2 + 1744*n + 1408)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*P[n - 1, x] - (441 + 1260*x - (336*n^2 + 336*n - 1136)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*D[P[n - 1, x], x])/(-336*n - 160*n*x + 128*n^3*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] = (-(840*n + 840 + (336*n^2 + 1744*n + 1408)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*v[i - 1] - (441 + 1260*x - (336*n^2 + 336*n - 1136)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*deriv(v[i - 1], x))/(-336*n - 160*n*x + 128*n^3*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-30cb3cad216ed189","steps":["basis A013611","\u03b8+1"],"edges":["e-21c167d0f859d82c","e-7b6fbbd5afd4edd0"],"text":"A207823 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A207823 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}