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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":"ec3ee7a7f67a5e0c4a2873be4cacda9eaee33fd34a7be6c212ddad8c066d48a0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d300994ed88c0a37","role":"sequence","name":"basis_map_A013611 of Coefficients of a special case of Poisson-Charlier polynomials.","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:      2,        8;\n   3:      5,       40,        48;\n   4:     16,      200,       528,        256;\n   5:     65,     1152,      5136,       7168,        1280;\n   6:    326,     7688,     50592,     130816,      120320,        6144;\n   7:   1957,    58504,    520512,    2040576,     3717120,     2586624,        28672;\n   8:  13700,   500200,   5670288,   30258176,    84491520,   119783424,     68210688,       131072;\n   9: 109601,  4749888,  65826720,  448002048,  1704728320,  3711811584,   4333772800,   2107637760,      589824;\n  10: 986410, 49634952, 816118848, 6785721856, 32901127680, 97505593344, 174603419648, 173946961920, 74130259968, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16)*P(n,x) = -(8*n^3-72*n^2+232*n-264+(96*n^3-960*n^2+3168*n-3456)*x+(384*n^3-4224*n^2+14208*n-14976)*x^2+(512*n^3-6144*n^2+20992*n-21504)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(16*n^3-136*n^2+400*n-408)*x+(96*n^3-864*n^2+2592*n-2592)*x^2+(256*n^3-2432*n^2+7424*n-7296)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*d/dx P(n-3,x) - (24*n^2-104*n+120+(192*n^2-832*n+896)*x+(384*n^2-1536*n+1536)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(12*n^3-84*n^2+192*n-136)*x+(48*n^3-384*n^2+944*n-736)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*d/dx P(n-2,x) + (16*n^2-48*n+40+(64*n^2-256*n+256)*x)*P(n-1,x) - (n^2-5*n+8+(8*n^2-48*n+72)*x+(16*n^2-112*n+160)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 8*x, P(3,x) = 5 + 40*x + 48*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[3, x_] := 5 + 40*x + 48*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*D[P[n - 3, x], x] - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*D[P[n - 2, x], x] + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; v[3] = 5 + 40*x + 48*x^2; for(n = 4, N, my(i = n); v[i] = (-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*deriv(v[i - 3], x) - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*deriv(v[i - 2], x) + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*v[i - 1] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*deriv(v[i - 1], x))/(8*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-c9dce8c25621fdea","steps":["basis A013611","\u03b8+1"],"edges":["e-5274dd88a300ad29","e-a6be8137c7581127"],"text":"A046716 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A046716 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}