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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":"4ac3945c075aff0f5c551df29d5089c146a9513783fdc6061d9d1aabfad50693","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-78b19c28b9f05152","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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,   2;\n  2: 1,  22,     3;\n  3: 1,  60,    90,      4;\n  4: 1, 116,   585,    232,       5;\n  5: 1, 190,  2055,   2740,     475,       6;\n  6: 1, 282,  5310,  15956,    8850,     846,      7;\n  7: 1, 392, 11403,  62132,   77665,   22806,   1372,      8;\n  8: 1, 520, 21630, 187712,  432575,  281568,  50470,   2080,    9;\n  9: 1, 666, 37530, 476616, 1781415, 2137698, 834078, 100080, 2997, 10;\n","recurrence_sentence":"The row polynomials satisfy -(6*n-6-(9*n^3-6*n^2+9*n-12)*x+(13*n^3-2*n^2+5*n-16)*x^2-(4*n^3-6*n^2+2*n)*x^3)*P(n,x) = -(6*n-6-(9*n^3-6*n^2-87*n+84)*x-(59*n^3+70*n^2+21*n-150)*x^2+(55*n^3+52*n^2-23*n-84)*x^3-(12*n^3-2*n^2-16*n+6)*x^4)*P(n-1,x) - (12*x-(18*n^2+38*n+112)*x^2+(134*n^2+274*n+204)*x^3-(74*n^2+94*n)*x^4+(8*n^2+8*n-4)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (9*n^3 - 6*n^2 - 87*n + 84)*x - (59*n^3 + 70*n^2 + 21*n - 150)*x^2 + (55*n^3 + 52*n^2 - 23*n - 84)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*P[n - 1, x] - (12*x - (18*n^2 + 38*n + 112)*x^2 + (134*n^2 + 274*n + 204)*x^3 - (74*n^2 + 94*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (9*n^3 - 6*n^2 + 9*n - 12)*x - (13*n^3 - 2*n^2 + 5*n - 16)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (9*n^3 - 6*n^2 - 87*n + 84)*x - (59*n^3 + 70*n^2 + 21*n - 150)*x^2 + (55*n^3 + 52*n^2 - 23*n - 84)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*v[i - 1] - (12*x - (18*n^2 + 38*n + 112)*x^2 + (134*n^2 + 274*n + 204)*x^3 - (74*n^2 + 94*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (9*n^3 - 6*n^2 + 9*n - 12)*x - (13*n^3 - 2*n^2 + 5*n - 16)*x^2 + (4*n^3 - 6*n^2 + 2*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-b30e7a5b4198051a","steps":["Veronese 2","\u03b8+1"],"edges":["e-864f7f13d69989fc","e-17ced6891870f1f4"],"text":"A272866 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A272866 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}