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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":"21bd51c368c27b81b45e710fe549b629ecba222bb87dd439d57ede8312e617cc","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-a6a633b989da1879","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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:        1;\n  1:        7,         20;\n  2:       49,        280,        225;\n  3:      343,       2940,       4725,        2000;\n  4:     2401,      27440,      66150,       56000,       15625;\n  5:    16807,     240100,     771750,      980000,      546875,      112500;\n  6:   117649,    2016840,    8103375,    13720000,    11484375,     4725000,      765625;\n  7:   823543,   16470860,   79413075,   168070000,   187578125,   115762500,    37515625,    5000000;\n  8:  5764801,  131766880,  741188700,  1882384000,  2626093750,  2160900000,  1050437500,  280000000,   31640625;\n  9: 40353607, 1037664180, 6670698300, 19765032000, 33088781250, 34034175000, 22059187500, 8820000000, 1993359375, 195312500;\n","recurrence_sentence":"The row polynomials satisfy (49+(105*n-35)*x+25*n^2*x^2)*P(n,x) = (343+(735*n+735)*x+(175*n^2+875*n+525)*x^2+(125*n^2+250*n+125)*x^3)*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[((343 + (735*n + 735)*x + (175*n^2 + 875*n + 525)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(49 + (105*n - 35)*x + 25*n^2*x^2)]];\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] = ((343 + (735*n + 735)*x + (175*n^2 + 875*n + 525)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(49 + (105*n - 35)*x + 25*n^2*x^2)); 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-a6a633b989da1879","steps":["(\u03b8+1)\u00b2"],"edges":["e-785d28f75272179a"],"text":"A038271 \u2192 (\u03b8+1)\u00b2"},"headline":"A038271 \u2192 (\u03b8+1)\u00b2","chain":null}