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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":"09e44a244bc0a3339d63a3cf252b4007750ef1c003c5f085271aa8460500a97d","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-d8e527df92e2a449","role":"sequence","name":"basis_map_A013610 of Riordan array (1+2x,x(1+x)).","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:   3,   12;\n  2:   4,   60,     81;\n  3:   7,  156,    567,     432;\n  4:  11,  348,   2106,    3888,    2025;\n  5:  18,  720,   6075,   18576,   22275,     8748;\n  6:  29, 1416,  15390,   66096,  129600,   113724,    35721;\n  7:  47, 2700,  35883,  199152,  548775,   778572,   535815,   139968;\n  8:  76, 5028,  79056,  538704, 1921725,  3822876,  4215078,  2379456,   531441;\n  9: 123, 9204, 167103, 1350864, 5929200, 15274008, 23540139, 21135168, 10097379, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (25+(225*n-210)*x+(540*n^2-900*n+324)*x^2+(405*n^3-594*n^2-513*n+270)*x^3+(81*n^4-891*n^2+162*n-81)*x^4-243*n^3*x^5)*P(n,x) = (25+(225*n+90)*x+(540*n^2+855*n+9)*x^2+(405*n^3+2241*n^2+54*n-540)*x^3+(81*n^4+1539*n^3+1458*n^2-2754*n-2025)*x^4+(243*n^4+729*n^3-1944*n^2-4617*n-2430)*x^5-(729*n^3+2187*n^2+2187*n+729)*x^6)*P(n-2,x) + (25+(225*n+90)*x+(540*n^2+540*n-306)*x^2+(405*n^3+1836*n^2-1917*n-2106)*x^3+(81*n^4+1377*n^3-243*n^2-6075*n-3807)*x^4+(243*n^4+243*n^3-3159*n^2-4860*n-1944)*x^5-(729*n^3+1458*n^2+729*n)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[((25 + (225*n + 90)*x + (540*n^2 + 855*n + 9)*x^2 + (405*n^3 + 2241*n^2 + 54*n - 540)*x^3 + (81*n^4 + 1539*n^3 + 1458*n^2 - 2754*n - 2025)*x^4 + (243*n^4 + 729*n^3 - 1944*n^2 - 4617*n - 2430)*x^5 - (729*n^3 + 2187*n^2 + 2187*n + 729)*x^6)*P[n - 2, x] + (25 + (225*n + 90)*x + (540*n^2 + 540*n - 306)*x^2 + (405*n^3 + 1836*n^2 - 1917*n - 2106)*x^3 + (81*n^4 + 1377*n^3 - 243*n^2 - 6075*n - 3807)*x^4 + (243*n^4 + 243*n^3 - 3159*n^2 - 4860*n - 1944)*x^5 - (729*n^3 + 1458*n^2 + 729*n)*x^6)*P[n - 1, x])/(25 + (225*n - 210)*x + (540*n^2 - 900*n + 324)*x^2 + (405*n^3 - 594*n^2 - 513*n + 270)*x^3 + (81*n^4 - 891*n^2 + 162*n - 81)*x^4 - 243*n^3*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((25 + (225*n + 90)*x + (540*n^2 + 855*n + 9)*x^2 + (405*n^3 + 2241*n^2 + 54*n - 540)*x^3 + (81*n^4 + 1539*n^3 + 1458*n^2 - 2754*n - 2025)*x^4 + (243*n^4 + 729*n^3 - 1944*n^2 - 4617*n - 2430)*x^5 - (729*n^3 + 2187*n^2 + 2187*n + 729)*x^6)*v[i - 2] + (25 + (225*n + 90)*x + (540*n^2 + 540*n - 306)*x^2 + (405*n^3 + 1836*n^2 - 1917*n - 2106)*x^3 + (81*n^4 + 1377*n^3 - 243*n^2 - 6075*n - 3807)*x^4 + (243*n^4 + 243*n^3 - 3159*n^2 - 4860*n - 1944)*x^5 - (729*n^3 + 1458*n^2 + 729*n)*x^6)*v[i - 1])/(25 + (225*n - 210)*x + (540*n^2 - 900*n + 324)*x^2 + (405*n^3 - 594*n^2 - 513*n + 270)*x^3 + (81*n^4 - 891*n^2 + 162*n - 81)*x^4 - 243*n^3*x^5)); 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-c27b11147fce8bfa","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-849840c29c946dc7","e-5b0a83b7507c56cc"],"text":"A113214 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A113214 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}