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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":"117768b86185c1ca4d523b5bbf41f2c861bc8e6b1700acf151032a04be76856d","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-97afe0d58b0705a8","role":"sequence","name":"derivative of One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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:   0;\n  1:   2;\n  2:  10,    12;\n  3:  28,   112,     36;\n  4:  60,   504,    540,      80;\n  5: 110,  1584,   3564,    1760,     150;\n  6: 182,  4004,  15444,   16016,    4550,     252;\n  7: 280,  8736,  51480,   91520,   54600,   10080,    392;\n  8: 408, 17136, 143208,  388960,  397800,  154224,  19992,   576;\n  9: 570, 31008, 348840, 1343680, 2099500, 1395360, 379848, 36480, 810;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1+(4*n^4-12*n^3+13*n^2-6*n+1)*x)*P(n,x) = (2*n^2+3*n+1+(4*n^4+7*n^2-14*n-1)*x+(12*n^4-16*n^3-5*n^2+3*n+4)*x^2)*P(n-1,x) + (4*n*x+(8*n^3-6*n-2)*x^2-(8*n^3-2*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*P[n - 1, x] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*v[i - 1] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x)); 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-2245b35ec7cd4857","steps":["D","\u03b8+1"],"edges":["e-2a66c334394f3697","e-909ce27bbce7c258"],"text":"A091044 \u2192 D \u2192 \u03b8+1"},"headline":"A091044 \u2192 D \u2192 \u03b8+1","chain":null}