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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":"6de62c926efdb74f5d85a55672123bbd520325dd05b9234a28f256dca07ecd30","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-a51ac9b78792f816","role":"sequence","name":"basis_map_A013610 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,             54;\n  2:          400,           2160,            2187;\n  3:         8000,          64800,          131220,           78732;\n  4:       160000,        1728000,         5248800,         6298560,          2657205;\n  5:      3200000,       43200000,       174960000,       314928000,        265720500,         86093442;\n  6:     64000000,     1036800000,      5248800000,     12597120000,      15943230000,      10331213040,       2711943423;\n  7:   1280000000,    24192000000,    146966400000,    440899200000,     744017400000,     723184912800,     379672079220,      83682825624;\n  8:  25600000000,   552960000000,   3919104000000,  14108774400000,   29760696000000,   38569862016000,   30373766337600,   13389252099840,   2541865828329;\n  9: 512000000000, 12441600000000, 100776960000000, 423263232000000, 1071385056000000, 1735643790720000, 1822425980256000, 1205032688985600, 457535849099220, 76255974849870;\n","recurrence_sentence":"The row polynomials satisfy (20+27*n*x)*P(n,x) = (400+(540*n+1080)*x+(729*n+729)*x^2)*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[((400 + (540*n + 1080)*x + (729*n + 729)*x^2)*P[n - 1, x])/(20 + 27*n*x)]];\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] = ((400 + (540*n + 1080)*x + (729*n + 729)*x^2)*v[i - 1])/(20 + 27*n*x)); 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 A013610","\u03b8+1"],"edges":["e-3bcf549fe20349b4","e-f2929144d93b6141"],"text":"A038323 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038323 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}