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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":"b9ef64e2a0e80b7374fbdeac6d20789025983c523d1691ebf258d80c2447b593","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-964d853754e5a7e8","role":"sequence","name":"reversal of A133080 * A007318.","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: 1,  4;\n  2: 1,  4,   3;\n  3: 1,  8,  15,   8;\n  4: 1,  8,  18,  16,   5;\n  5: 1, 12,  42,  64,  45,   12;\n  6: 1, 12,  45,  80,  75,   36,   7;\n  7: 1, 16,  81, 200, 275,  216,  91,  16;\n  8: 1, 16,  84, 224, 350,  336, 196,  64,   9;\n  9: 1, 20, 132, 448, 910, 1176, 980, 512, 153, 20;\n","recurrence_sentence":"The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2+10*n*x+(9*n^2+17*n-18)*x^2+(2*n^3+15*n^2-7*n-10)*x^3+(3*n^3-3*n)*x^4)*P(n-1,x) - (4*x+(6*n+15)*x^2+(2*n^2+18*n+16)*x^3+(5*n^2+15*n+5)*x^4+(3*n^2+3*n)*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[((2 + 10*n*x + (9*n^2 + 17*n - 18)*x^2 + (2*n^3 + 15*n^2 - 7*n - 10)*x^3 + (3*n^3 - 3*n)*x^4)*P[n - 1, x] - (4*x + (6*n + 15)*x^2 + (2*n^2 + 18*n + 16)*x^3 + (5*n^2 + 15*n + 5)*x^4 + (3*n^2 + 3*n)*x^5)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*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] = ((2 + 10*n*x + (9*n^2 + 17*n - 18)*x^2 + (2*n^3 + 15*n^2 - 7*n - 10)*x^3 + (3*n^3 - 3*n)*x^4)*v[i - 1] - (4*x + (6*n + 15)*x^2 + (2*n^2 + 18*n + 16)*x^3 + (5*n^2 + 15*n + 5)*x^4 + (3*n^2 + 3*n)*x^5)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*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-d75669a8cd7f9393","steps":["reverse","\u03b8+1"],"edges":["e-7d9675c994dd8650","e-faec78fde58eb33c"],"text":"A133087 \u2192 reverse \u2192 \u03b8+1"},"headline":"A133087 \u2192 reverse \u2192 \u03b8+1","chain":null}