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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":"06d98fe1daf94ab38e35d14ff51881ab9b34a6fa3318c8ca7ce90126d5223a4d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1eee05e015d7347b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,       12;\n   3:      5,       60,        81;\n   4:     16,      276,       729,       432;\n   5:     65,     1392,      5265,      6048,       2025;\n   6:    326,     8028,     38070,     62640,      40500,       8748;\n   7:   1957,    53148,    294354,    611280,     567000,     236196,     35721;\n   8:  13700,   400500,   2485161,   6084288,    7158375,    4286520,   1250235,    139968;\n   9: 109601,  3396864,  23009346,  63733824,   88976475,   67849488,  28505358,   6158592,   531441;\n  10: 986410, 32079852, 233141004, 711380448, 1131250050, 1028581092, 552103776, 172160640, 28697814, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(12*n-36)*x)*P(n-3,x) - (12*n-36)*x^2*d/dx P(n-3,x) - (n^2-4+(12*n-12)*x-81*x^2)*P(n-2,x) + ((12*n-12)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n-1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 12*x, P(3,x) = 5 + 60*x + 81*x^2, P(4,x) = 16 + 276*x + 729*x^2 + 432*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 12*x;\nP[3, x_] := 5 + 60*x + 81*x^2;\nP[4, x_] := 16 + 276*x + 729*x^2 + 432*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (12*n - 36)*x)*P[n - 3, x] - (12*n - 36)*x^2*D[P[n - 3, x], x] - (n^2 - 4 + (12*n - 12)*x - 81*x^2)*P[n - 2, x] + ((12*n - 12)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 12*x; v[3] = 5 + 60*x + 81*x^2; v[4] = 16 + 276*x + 729*x^2 + 432*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (12*n - 36)*x)*v[i - 3] - (12*n - 36)*x^2*deriv(v[i - 3], x) - (n^2 - 4 + (12*n - 12)*x - 81*x^2)*v[i - 2] + ((12*n - 12)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-792aa57b19fb3b04","e-cbe36ebc0710a263"],"text":"A094816 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A094816 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}