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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":"3493537751bdb3ba08c10cb00a060898061c5dd189f22ea695c797eaa1cf21aa","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-226ddb8fc3536584","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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,    6;\n  2:  3,   24,    27;\n  3:  5,   60,   162,    108;\n  4:  8,  132,   567,    864,    405;\n  5: 13,  270,  1593,   3888,   4050,    1458;\n  6: 21,  528,  3969,  13392,  22275,   17496,    5103;\n  7: 34, 1002,  9153,  39528,  91125,  113724,   71442,   17496;\n  8: 55, 1860, 20007, 105408, 311850,  539460,  535815,  279936,   59049;\n  9: 89, 3396, 42039, 261576, 946485, 2103894, 2893401, 2379456, 1062882, 196830;\n","recurrence_sentence":"The row polynomials satisfy (30*n+30+(45*n^2+117*n+72)*x+(27*n^3+54*n^2+27*n)*x^2-(81*n^2+81*n)*x^3)*P(n,x) = -(60*n-180+(135*n^2-351*n-612)*x+(54*n^3-108*n^2-972*n-594)*x^2-(405*n^2+243*n-324)*x^3+(486*n+486)*x^4)*P(n-1,x) + (25+(75*n+135)*x+(45*n^2+315*n+162)*x^2+(162*n^2+162*n-216)*x^3+(81*n^2-405*n-567)*x^4-(243*n+243)*x^5)*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[(-(60*n - 180 + (135*n^2 - 351*n - 612)*x + (54*n^3 - 108*n^2 - 972*n - 594)*x^2 - (405*n^2 + 243*n - 324)*x^3 + (486*n + 486)*x^4)*P[n - 1, x] + (25 + (75*n + 135)*x + (45*n^2 + 315*n + 162)*x^2 + (162*n^2 + 162*n - 216)*x^3 + (81*n^2 - 405*n - 567)*x^4 - (243*n + 243)*x^5)*D[P[n - 1, x], x])/(30*n + 30 + (45*n^2 + 117*n + 72)*x + (27*n^3 + 54*n^2 + 27*n)*x^2 - (81*n^2 + 81*n)*x^3)]];\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] = (-(60*n - 180 + (135*n^2 - 351*n - 612)*x + (54*n^3 - 108*n^2 - 972*n - 594)*x^2 - (405*n^2 + 243*n - 324)*x^3 + (486*n + 486)*x^4)*v[i - 1] + (25 + (75*n + 135)*x + (45*n^2 + 315*n + 162)*x^2 + (162*n^2 + 162*n - 216)*x^3 + (81*n^2 - 405*n - 567)*x^4 - (243*n + 243)*x^5)*deriv(v[i - 1], x))/(30*n + 30 + (45*n^2 + 117*n + 72)*x + (27*n^3 + 54*n^2 + 27*n)*x^2 - (81*n^2 + 81*n)*x^3)); 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-21854f4257effb01","steps":["basis A013610","\u03b8+1"],"edges":["e-b1c4a0832f5f9447","e-992c8706dbc7f64e"],"text":"A030528 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A030528 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}