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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":"cccb75b26c290514fa41d3cc7a70b7bdd94a8331819cc5ea32b272ba6d4e8cd3","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-2fe6a26f73f75da6","role":"sequence","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    9,     48,      36;\n  1:   18,    168,     288,      128;\n  2:   36,    480,    1332,     1280,      400;\n  3:   72,   1248,    4824,     7296,     4800,     1152;\n  4:  144,   3072,   15264,    31744,    32400,    16128,     3136;\n  5:  288,   7296,   44352,   117760,   164000,   125568,    50176,     8192;\n  6:  576,  16896,  121536,   393216,   696000,   723456,   442176,   147456,    20736;\n  7: 1152,  38400,  319104,  1218560,  2620800,  3451392,  2853760,  1449984,   414720,    51200;\n  8: 2304,  86016,  811008,  3571712,  9049600, 14450688, 15102976, 10354688,  4499712,  1126400,  123904;\n  9: 4608, 190464, 2009088, 10027008, 29260800, 54964224, 69543936, 60162048, 35209728, 13363200, 2973696, 294912;\n","recurrence_sentence":"The row polynomials satisfy (9+(27*n+39)*x+(9*n^2+60*n+72)*x^2+(12*n^2+52*n+56)*x^3+(4*n^2+16*n+16)*x^4)*P(n,x) = (18+(54*n+150)*x+(18*n^2+210*n+414)*x^2+(42*n^2+308*n+522)*x^3+(32*n^2+200*n+312)*x^4+(8*n^2+48*n+72)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 48*x + 36*x^2.","mathematica":"P[0, x_] := 9 + 48*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((18 + (54*n + 150)*x + (18*n^2 + 210*n + 414)*x^2 + (42*n^2 + 308*n + 522)*x^3 + (32*n^2 + 200*n + 312)*x^4 + (8*n^2 + 48*n + 72)*x^5)*P[n - 1, x])/(9 + (27*n + 39)*x + (9*n^2 + 60*n + 72)*x^2 + (12*n^2 + 52*n + 56)*x^3 + (4*n^2 + 16*n + 16)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 48*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + (54*n + 150)*x + (18*n^2 + 210*n + 414)*x^2 + (42*n^2 + 308*n + 522)*x^3 + (32*n^2 + 200*n + 312)*x^4 + (8*n^2 + 48*n + 72)*x^5)*v[i - 1])/(9 + (27*n + 39)*x + (9*n^2 + 60*n + 72)*x^2 + (12*n^2 + 52*n + 56)*x^3 + (4*n^2 + 16*n + 16)*x^4)); 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-f48bed27dbc29c5f","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-8866dbf598c953fc","e-25751e82c60cc375"],"text":"A391533 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A391533 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}