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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":"c63e6ce3980790064b32ef5963dbc114708169a247024195973e8b69a6981378","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-e7c079f81f452bf3","role":"sequence","name":"Another version of Fibonacci-Pascal triangle A037027.","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: 0, 2;\n  2: 0, 2, 6;\n  3: 0, 0, 6, 12;\n  4: 0, 0, 3, 20, 25;\n  5: 0, 0, 0, 12, 50,  48;\n  6: 0, 0, 0,  4, 45, 120,  91;\n  7: 0, 0, 0,  0, 20, 132, 266, 168;\n  8: 0, 0, 0,  0,  5,  84, 357, 568,  306;\n  9: 0, 0, 0,  0,  0,  30, 280, 888, 1170, 550;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n+(4*n^3+2*n^2)*x+(5*n^3-5*n^2)*x^2)*P(n,x) = -(2*n^3+4*n^2-2*n-4+(14*n^3+22*n^2-20*n-28)*x+(39*n^3+46*n^2-53*n-44)*x^2+(50*n^3+30*n^2-60*n-20)*x^3+(25*n^3-25*n)*x^4)*P(n-1,x) + ((4*n^2+12*n+8)*x+(25*n^2+59*n+26)*x^2+(61*n^2+107*n+28)*x^3+(65*n^2+85*n+10)*x^4+(25*n^2+25*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*n^3 + 4*n^2 - 2*n - 4 + (14*n^3 + 22*n^2 - 20*n - 28)*x + (39*n^3 + 46*n^2 - 53*n - 44)*x^2 + (50*n^3 + 30*n^2 - 60*n - 20)*x^3 + (25*n^3 - 25*n)*x^4)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (25*n^2 + 59*n + 26)*x^2 + (61*n^2 + 107*n + 28)*x^3 + (65*n^2 + 85*n + 10)*x^4 + (25*n^2 + 25*n)*x^5)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (4*n^3 + 2*n^2)*x + (5*n^3 - 5*n^2)*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*n^3 + 4*n^2 - 2*n - 4 + (14*n^3 + 22*n^2 - 20*n - 28)*x + (39*n^3 + 46*n^2 - 53*n - 44)*x^2 + (50*n^3 + 30*n^2 - 60*n - 20)*x^3 + (25*n^3 - 25*n)*x^4)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (25*n^2 + 59*n + 26)*x^2 + (61*n^2 + 107*n + 28)*x^3 + (65*n^2 + 85*n + 10)*x^4 + (25*n^2 + 25*n)*x^5)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (4*n^3 + 2*n^2)*x + (5*n^3 - 5*n^2)*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-e7c079f81f452bf3","steps":["\u03b8+1"],"edges":["e-46c0275dfe2e49c4"],"text":"A111006 \u2192 \u03b8+1"},"headline":"A111006 \u2192 \u03b8+1","chain":null}