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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":"9bb832e5fcf91acc00352560f3986faefd34a0fec84762caca59ef27cfb38865","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-b4e192fec3e320c1","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+6x)^n.","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:        7,         36;\n  2:       49,        504,         972;\n  3:      343,       5292,       20412,        23328;\n  4:     2401,      49392,      285768,       653184,        524880;\n  5:    16807,     432180,     3333960,     11430720,      18370800,      11337408;\n  6:   117649,    3630312,    35006580,    160030080,     385786800,     476171136,     238085568;\n  7:   823543,   29647548,   343064484,   1960368480,    6301184400,   11666192832,   11666192832,    4897760256;\n  8:  5764801,  237180384,  3201935184,  21956126976,   88216581600,  217768932864,  326653399296,  274274574336,   99179645184;\n  9: 40353607, 1867795524, 28817416656, 230539333248, 1111528928160, 3429860692608, 6859721385216, 8639649091584, 6248317646592, 1983592903680;\n","recurrence_sentence":"The row polynomials satisfy (7+18*n*x)*P(n,x) = (49+(126*n+252)*x+(324*n+324)*x^2)*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[((49 + (126*n + 252)*x + (324*n + 324)*x^2)*P[n - 1, x])/(7 + 18*n*x)]];\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] = ((49 + (126*n + 252)*x + (324*n + 324)*x^2)*v[i - 1])/(7 + 18*n*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-b7cf68162d40a956","steps":["x \u21a6 x/(x+1)","\u03b8+1"],"edges":["e-149dcf3d57e14e87","e-48ad8211afef4938"],"text":"A038277 \u2192 x \u21a6 x/(x+1) \u2192 \u03b8+1"},"headline":"A038277 \u2192 x \u21a6 x/(x+1) \u2192 \u03b8+1","chain":null}