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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":"d31a7d153c72c2e0f8921ae586bfa2f0419b68cc2af55ec732c5756fc4a6f9b2","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-dd4c9e25ea6d895a","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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:         2,          6;\n  2:         8,         36,         27;\n  3:        48,        264,        324,        108;\n  4:       384,       2400,       3780,       2160,        405;\n  5:      3840,      26304,      48600,      36720,      12150,       1458;\n  6:     46080,     338688,     701568,     635040,     283500,      61236,      5103;\n  7:    645120,    5018112,   11346048,   11696832,    6350400,    1877904,    285768,    17496;\n  8:  10321920,   84160512,  204118272,  232533504,  145469520,   52907904,  11144952,  1259712,   59049;\n  9: 185794560, 1576820736, 4052851200, 5002076160, 3490452000, 1476032544, 385786800, 60886080, 5314410, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(12*n-12)*x+9*x^2)*P(n-2,x) + (4*n-2+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*P[n - 2, x] + (4*n - 2 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*v[i - 2] + (4*n - 2 + 6*x)*v[i - 1]); 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-4f6e747b794bcc2e","steps":["basis A013610","\u03b8+1"],"edges":["e-45a2b063eb896a2b","e-29057f910d9e6953"],"text":"A028338 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A028338 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}