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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":"1fd5e28b6c74e2ceb50053cf8e9086a0d3d48eb5de89319356916d1a2a6629df","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-3e465972400fd143","role":"sequence","name":"shift_x_plus_one of Triangular array of Motzkin polynomial coefficients.","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;\n  2:   2,    2;\n  3:   4,    6;\n  4:   9,   20,    6;\n  5:  21,   60,   30;\n  6:  51,  180,  135,   20;\n  7: 127,  532,  525,  140;\n  8: 323, 1568, 1932,  784,  70;\n  9: 835, 4608, 6804, 3696, 630;\n","recurrence_sentence":"The row polynomials satisfy (6*n+12+(4*n^2+14*n+12)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (30*n-12+(26*n^2+52*n-48)*x+(5*n^3+35*n^2+14*n-48)*x^2+(4*n^3+8*n^2-4*n-8)*x^3)*P(n-1,x) - (18+(24*n+78)*x+(6*n^2+74*n+120)*x^2+(14*n^2+74*n+76)*x^3+(8*n^2+24*n+16)*x^4)*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[((30*n - 12 + (26*n^2 + 52*n - 48)*x + (5*n^3 + 35*n^2 + 14*n - 48)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*P[n - 1, x] - (18 + (24*n + 78)*x + (6*n^2 + 74*n + 120)*x^2 + (14*n^2 + 74*n + 76)*x^3 + (8*n^2 + 24*n + 16)*x^4)*D[P[n - 1, x], x])/(6*n + 12 + (4*n^2 + 14*n + 12)*x + (n^3 + 3*n^2 + 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] = ((30*n - 12 + (26*n^2 + 52*n - 48)*x + (5*n^3 + 35*n^2 + 14*n - 48)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*v[i - 1] - (18 + (24*n + 78)*x + (6*n^2 + 74*n + 120)*x^2 + (14*n^2 + 74*n + 76)*x^3 + (8*n^2 + 24*n + 16)*x^4)*deriv(v[i - 1], x))/(6*n + 12 + (4*n^2 + 14*n + 12)*x + (n^3 + 3*n^2 + 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-169bd483754ab390","steps":["basis A007318","\u03b8+1"],"edges":["e-5fd319f73e442e4d","e-44174138345282b8"],"text":"A055151 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A055151 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}