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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":"aa1bacff7a917671946c60f05a4aab2e74c7de23cbd2c8ff9c2027aa71837326","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8d5fed5e270db201","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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;\n  2:    4,    18;\n  3:    7,    36;\n  4:   19,   162,    243;\n  5:   40,   396,    729;\n  6:   97,  1224,   3645,   2916;\n  7:  217,  3132,  11178,  11664;\n  8:  508,  8550,  38637,  64152,  32805;\n  9: 1159, 21852, 114453, 233280, 164025;\n","recurrence_sentence":"The row polynomials satisfy (78*n+(117*n^2+171*n)*x+(81*n^3+81*n^2)*x^2)*P(n,x) = (546*n-468+(1170*n^2+2250*n-3132)*x+(567*n^3+4293*n^2+1458*n-6156)*x^2+(1458*n^3+2916*n^2-1458*n-2916)*x^3)*P(n-1,x) - (169+(507*n+1716)*x+(351*n^2+3978*n+6381)*x^2+(2025*n^2+10287*n+10206)*x^3+(2916*n^2+8748*n+5832)*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[((546*n - 468 + (1170*n^2 + 2250*n - 3132)*x + (567*n^3 + 4293*n^2 + 1458*n - 6156)*x^2 + (1458*n^3 + 2916*n^2 - 1458*n - 2916)*x^3)*P[n - 1, x] - (169 + (507*n + 1716)*x + (351*n^2 + 3978*n + 6381)*x^2 + (2025*n^2 + 10287*n + 10206)*x^3 + (2916*n^2 + 8748*n + 5832)*x^4)*D[P[n - 1, x], x])/(78*n + (117*n^2 + 171*n)*x + (81*n^3 + 81*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] = ((546*n - 468 + (1170*n^2 + 2250*n - 3132)*x + (567*n^3 + 4293*n^2 + 1458*n - 6156)*x^2 + (1458*n^3 + 2916*n^2 - 1458*n - 2916)*x^3)*v[i - 1] - (169 + (507*n + 1716)*x + (351*n^2 + 3978*n + 6381)*x^2 + (2025*n^2 + 10287*n + 10206)*x^3 + (2916*n^2 + 8748*n + 5832)*x^4)*deriv(v[i - 1], x))/(78*n + (117*n^2 + 171*n)*x + (81*n^3 + 81*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-e189038e95d51606","steps":["basis A013610","\u03b8+1"],"edges":["e-99e9156f8de29382","e-2d4026f9c6fde8e8"],"text":"A304236 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A304236 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}