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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":"5485840ddae5f3845eaefc71c05549b4205350f0f02a9305baf140a192bbc980","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-76276d62adb9bc90","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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:           20,             72;\n  2:          400,           2880,            2916;\n  3:         8000,          86400,          174960,           93312;\n  4:       160000,        2304000,         6998400,         7464960,          2624400;\n  5:      3200000,       57600000,       233280000,       373248000,        262440000,         68024448;\n  6:     64000000,     1382400000,      6998400000,     14929920000,      15746400000,       8162933760,       1666598976;\n  7:   1280000000,    32256000000,    195955200000,    522547200000,     734832000000,     571405363200,     233323856640,     39182082048;\n  8:  25600000000,   737280000000,   5225472000000,  16721510400000,   29393280000000,   30474952704000,   18665908531200,   6269133127680,    892616806656;\n  9: 512000000000, 16588800000000, 134369280000000, 501645312000000, 1058158080000000, 1371372871680000, 1119954511872000, 564221981491200, 160671025198080, 19835929036800;\n","recurrence_sentence":"The row polynomials satisfy (100+(270*n-90)*x+81*n^2*x^2)*P(n,x) = (2000+(5400*n+5400)*x+(1620*n^2+8100*n+4860)*x^2+(1458*n^2+2916*n+1458)*x^3)*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[((2000 + (5400*n + 5400)*x + (1620*n^2 + 8100*n + 4860)*x^2 + (1458*n^2 + 2916*n + 1458)*x^3)*P[n - 1, x])/(100 + (270*n - 90)*x + 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] = ((2000 + (5400*n + 5400)*x + (1620*n^2 + 8100*n + 4860)*x^2 + (1458*n^2 + 2916*n + 1458)*x^3)*v[i - 1])/(100 + (270*n - 90)*x + 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-4e096b5929fd3f2e","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-66940ce826511afb","e-50715fffcfa15fac"],"text":"A038323 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038323 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}