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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":"15a9284654447487481b268ef7029356f78499ebf8177af1074bce5933bb1e39","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-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.","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:   2,         8;\n  2:   4,       112,          36;\n  3:   8,       976,        2196,          128;\n  4:  16,      7376,       52704,        29504,           400;\n  5:  32,     53040,      914580,      1625920,        331500,          1152;\n  6:  64,    374736,    13600440,     55281280,      37779000,       3372624,          3136;\n  7: 128,   2631344,   185970492,   1471710080,    2299547000,     743881968,      32233964,         8192;\n  8: 256,  18438352,  2421102672,  33945654976,  101180068000,   76377723696,   13181558992,    295013632,      20736;\n  9: 512, 129111472, 30588224868, 714809975488, 3658043815300, 5267583094032, 2189105549932, 217516265728, 2614507308, 51200;\n","recurrence_sentence":"The row polynomials satisfy (25*n^2-30*n+9)*P(n,x) = (150*n^2-30*n-36-(1000*n^3-1800*n^2-1420*n+2784)*x+(5625*n^4-14750*n^3+10025*n^2-960*n-576)*x^2)*P(n-2,x) - ((500*n^2-100*n-645)*x-(1000*n^3+1700*n^2+5*n-2706)*x^2-(2500*n^4-14750*n^3+13025*n^2-1860*n-711)*x^3)*d/dx P(n-2,x) - ((625*n^2-125*n-525)*x^2-(2500*n^3-2000*n^2+325*n-390)*x^3+(2500*n^3-2625*n^2+450*n+135)*x^4)*d^2/dx^2 P(n-2,x) - (25*n^2+45*n-36-(500*n^3-900*n^2+540*n-108)*x)*P(n-1,x) + ((250*n^2-175*n-60)*x-(500*n^3-400*n^2-60*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*P[n - 2, x] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*D[P[n - 2, x], x] - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*P[n - 1, x] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*D[P[n - 1, x], x])/(25*n^2 - 30*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*v[i - 2] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*deriv(v[i - 2], x) - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*v[i - 1] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*deriv(v[i - 1], x))/(25*n^2 - 30*n + 9)); 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-ee6acb764200c6c7","steps":["(\u03b8+1)\u00b2"],"edges":["e-d8445ca60a65b517"],"text":"A257614 \u2192 (\u03b8+1)\u00b2"},"headline":"A257614 \u2192 (\u03b8+1)\u00b2","chain":null}