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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":"be4aa686001131820ea7a302bf75ede9e546c9f81fbc03cf01557e7e9b70ff23","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-f074ae9ffad82cf4","role":"sequence","name":"basis_map_A013609 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,    12;\n  3:    7,    24;\n  4:   19,   108,   108;\n  5:   40,   264,   324;\n  6:   97,   816,  1620,   864;\n  7:  217,  2088,  4968,  3456;\n  8:  508,  5700, 17172, 19008,  6480;\n  9: 1159, 14568, 50868, 69120, 32400;\n","recurrence_sentence":"The row polynomials satisfy (52*n+(52*n^2+76*n)*x+(24*n^3+24*n^2)*x^2)*P(n,x) = (364*n-312+(520*n^2+1000*n-1392)*x+(168*n^3+1272*n^2+432*n-1824)*x^2+(288*n^3+576*n^2-288*n-576)*x^3)*P(n-1,x) - (169+(338*n+1144)*x+(156*n^2+1768*n+2836)*x^2+(600*n^2+3048*n+3024)*x^3+(576*n^2+1728*n+1152)*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[((364*n - 312 + (520*n^2 + 1000*n - 1392)*x + (168*n^3 + 1272*n^2 + 432*n - 1824)*x^2 + (288*n^3 + 576*n^2 - 288*n - 576)*x^3)*P[n - 1, x] - (169 + (338*n + 1144)*x + (156*n^2 + 1768*n + 2836)*x^2 + (600*n^2 + 3048*n + 3024)*x^3 + (576*n^2 + 1728*n + 1152)*x^4)*D[P[n - 1, x], x])/(52*n + (52*n^2 + 76*n)*x + (24*n^3 + 24*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] = ((364*n - 312 + (520*n^2 + 1000*n - 1392)*x + (168*n^3 + 1272*n^2 + 432*n - 1824)*x^2 + (288*n^3 + 576*n^2 - 288*n - 576)*x^3)*v[i - 1] - (169 + (338*n + 1144)*x + (156*n^2 + 1768*n + 2836)*x^2 + (600*n^2 + 3048*n + 3024)*x^3 + (576*n^2 + 1728*n + 1152)*x^4)*deriv(v[i - 1], x))/(52*n + (52*n^2 + 76*n)*x + (24*n^3 + 24*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 A013609","\u03b8+1"],"edges":["e-79f6722e524950c5","e-6d6195c831ce4bc3"],"text":"A304236 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A304236 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}