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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":"6e2f693b4b2e5466008e228e2256d83279b5da8f5ce282c6062f49a56c124dbc","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-1397bdceeefc7b3d","role":"sequence","name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*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:       3,         8;\n  2:      15,        80,        54;\n  3:      84,       672,       945,        320;\n  4:     495,      5280,     11340,       8064,       1750;\n  5:    3003,     40040,    115830,     126720,      57750,       9072;\n  6:   18564,    297024,   1081080,    1601600,    1126125,     370656,      45276;\n  7:  116280,   2170560,   9523332,   17821440,   17017000,    8648640,    2207205,    219648;\n  8:  735471,  15690048,  80582040,  182327040,  220447500,  152373312,   60035976,  12446720,  1042470;\n  9: 4686825, 112483800, 661923900, 1757285376, 2574148500, 2256297120, 1209815880, 386979840, 67343562, 4862000;\n","recurrence_sentence":"The row polynomials satisfy (36*n^4-42*n^3+16*n^2-2*n+(72*n^5-124*n^4+76*n^3-20*n^2+2*n)*x+(16*n^6-24*n^5+12*n^4-2*n^3)*x^2)*P(n,x) = (243*n^4-405*n^3+243*n^2-63*n+6+(486*n^5-216*n^4-594*n^3+561*n^2-173*n+18)*x+(108*n^6+504*n^5-696*n^4-132*n^3+411*n^2-159*n+18)*x^2+(96*n^6+50*n^5-212*n^4+36*n^3+85*n^2-45*n+6)*x^3+(4*n^6-3*n^4-n^3)*x^4)*P(n-1,x) - ((81*n^3-81*n^2+27*n-3)*x+(162*n^4+54*n^3-171*n^2+73*n-9)*x^2+(36*n^5+228*n^4-64*n^3-113*n^2+65*n-9)*x^3+(40*n^5+70*n^4-36*n^3-23*n^2+19*n-3)*x^4+(4*n^5+4*n^4+n^3)*x^5)*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[((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (486*n^5 - 216*n^4 - 594*n^3 + 561*n^2 - 173*n + 18)*x + (108*n^6 + 504*n^5 - 696*n^4 - 132*n^3 + 411*n^2 - 159*n + 18)*x^2 + (96*n^6 + 50*n^5 - 212*n^4 + 36*n^3 + 85*n^2 - 45*n + 6)*x^3 + (4*n^6 - 3*n^4 - n^3)*x^4)*P[n - 1, x] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (162*n^4 + 54*n^3 - 171*n^2 + 73*n - 9)*x^2 + (36*n^5 + 228*n^4 - 64*n^3 - 113*n^2 + 65*n - 9)*x^3 + (40*n^5 + 70*n^4 - 36*n^3 - 23*n^2 + 19*n - 3)*x^4 + (4*n^5 + 4*n^4 + n^3)*x^5)*D[P[n - 1, x], x])/(36*n^4 - 42*n^3 + 16*n^2 - 2*n + (72*n^5 - 124*n^4 + 76*n^3 - 20*n^2 + 2*n)*x + (16*n^6 - 24*n^5 + 12*n^4 - 2*n^3)*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] = ((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (486*n^5 - 216*n^4 - 594*n^3 + 561*n^2 - 173*n + 18)*x + (108*n^6 + 504*n^5 - 696*n^4 - 132*n^3 + 411*n^2 - 159*n + 18)*x^2 + (96*n^6 + 50*n^5 - 212*n^4 + 36*n^3 + 85*n^2 - 45*n + 6)*x^3 + (4*n^6 - 3*n^4 - n^3)*x^4)*v[i - 1] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (162*n^4 + 54*n^3 - 171*n^2 + 73*n - 9)*x^2 + (36*n^5 + 228*n^4 - 64*n^3 - 113*n^2 + 65*n - 9)*x^3 + (40*n^5 + 70*n^4 - 36*n^3 - 23*n^2 + 19*n - 3)*x^4 + (4*n^5 + 4*n^4 + n^3)*x^5)*deriv(v[i - 1], x))/(36*n^4 - 42*n^3 + 16*n^2 - 2*n + (72*n^5 - 124*n^4 + 76*n^3 - 20*n^2 + 2*n)*x + (16*n^6 - 24*n^5 + 12*n^4 - 2*n^3)*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-1397bdceeefc7b3d","steps":["(\u03b8+1)\u00b2"],"edges":["e-d98c702421f504c3"],"text":"A357613 \u2192 (\u03b8+1)\u00b2"},"headline":"A357613 \u2192 (\u03b8+1)\u00b2","chain":null}