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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":"3b5e984ee46f993c71982b94aae84c61194969e8cc92e9cae51c2713f8254011","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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:          144,           336,            147;\n  1:         1728,          6048,           5292,           1372;\n  2:        20736,         96768,         127008,          65856,           12005;\n  3:       248832,       1451520,        2540160,        1975680,          720300,          100842;\n  4:      2985984,      20901888,       45722880,       47416320,        25930800,         7260624,         823543;\n  5:     35831808,     292626432,      768144384,      995742720,       726062400,       304946208,       69177612,        6588344;\n  6:    429981696,    4013162496,    12290310144,    19118260224,     17425497600,      9758278656,     3320525376,      632481024,       51883209;\n  7:   5159780352,   54177693696,   189621927936,   344128684032,    376390748160,    263473523712,   119538913536,    34153975296,     5603386572,     403536070;\n  8:  61917364224,  722369249280,  2844328919040,  5899348869120,   7527814963200,   6323364569088,  3586167406080,  1366159011840,   336203194320,   48424328400,   3107227739;\n  9: 743008370688, 9535274090496, 41716824145920, 97339256340480, 141953082163200, 139114020519936, 94674819520512, 45083247390720, 14792940550080, 3196005674400, 410154061548, 23727920916;\n","recurrence_sentence":"The row polynomials satisfy (12+(7*n+14)*x)*P(n,x) = (144+(84*n+336)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 336*x + 147*x^2.","mathematica":"P[0, x_] := 144 + 336*x + 147*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((144 + (84*n + 336)*x + (49*n + 147)*x^2)*P[n - 1, x])/(12 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 336*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (84*n + 336)*x + (49*n + 147)*x^2)*v[i - 1])/(12 + (7*n + 14)*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-5cf6acfdac607d6d","steps":["\u03b8+1"],"edges":["e-7063d9c027e1d23f"],"text":"A038333 \u2192 \u03b8+1"},"headline":"A038333 \u2192 \u03b8+1","chain":null}