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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":"73526ebc4d3878d8088518393e081cf08f9accfbee14736634134da145b0b424","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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,    6;\n  2: 1,   24,      45;\n  3: 1,   66,     423,      420;\n  4: 1,  156,    2358,     7320,       4725;\n  5: 1,  342,   10350,    70152,     133425,       62370;\n  6: 1,  720,   39771,   501408,    1982475,     2615760,      945945;\n  7: 1, 1482,  140913,  3013164,   21194055,    56137050,    55449135,    16216200;\n  8: 1, 3012,  473868, 16186104,  185065830,   845479620,  1632515220,  1270397520,   310134825;\n  9: 1, 6078, 1539072, 80540016, 1410912270, 10180694628, 33128250120, 49343545440, 31375157625, 6547290750;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-6*n-3)*x^2*P(n-2,x) - ((3*n+1)*x^2-(3*n+1)*x^3)*d/dx P(n-2,x) + (1+6*n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 6*n - 3)*x^2*P[n - 2, x] - ((3*n + 1)*x^2 - (3*n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 6*n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 6*n - 3)*x^2*v[i - 2] - ((3*n + 1)*x^2 - (3*n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 6*n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 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-86fb6463ff8368f2","steps":["\u03b8+1"],"edges":["e-a043eb561638fafd"],"text":"A219512 \u2192 \u03b8+1"},"headline":"A219512 \u2192 \u03b8+1","chain":null}