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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":"e69168858ec1cbecfba5b903fce35fcaa1988aeebcd33d2a3fe2934401ebfc01","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-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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:          1,           8;\n  2:          2,          24,           36;\n  3:         10,         136,          288,          128;\n  4:        100,        1440,         3492,         2304,          400;\n  5:       1700,       25280,        65844,        51584,        14000,        1152;\n  6:      44200,      670880,      1825704,      1575296,       525200,       70272,       3136;\n  7:    1635400,    25176160,     70570008,     64777344,     24355200,     4112640,     307328,      8192;\n  8:   81770000,  1271891200,   3641793120,   3489782784,   1420189200,   275774976,   26561920,   1212416,   20736;\n  9: 5315050000, 83327088000, 242440063200, 239784478720, 103217869200, 22015518336, 2477245568, 148193280, 4416768, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(8*n^2-32*n+40)*x-36*x^2)*P(n-2,x) + ((8*n^2-32*n+40)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+7+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*P[n - 2, x] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 7 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*v[i - 2] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 7 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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-54284d24a094e2d3","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-327e7f94e2e44f03","e-984fcf2114857f29"],"text":"A269944 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A269944 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}