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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":"bd1895dc1f4dd45c34410b5bb4ccc2a73cb5502d607c7332a02c166762c738b4","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,        10;\n  2:       49,       140,         75;\n  3:      343,      1470,       1575,        500;\n  4:     2401,     13720,      22050,      14000,       3125;\n  5:    16807,    120050,     257250,     245000,     109375,      18750;\n  6:   117649,   1008420,    2701125,    3430000,    2296875,     787500,     109375;\n  7:   823543,   8235430,   26471025,   42017500,   37515625,   19293750,    5359375,     625000;\n  8:  5764801,  65883440,  247062900,  470596000,  525218750,  360150000,  150062500,   35000000,   3515625;\n  9: 40353607, 518832090, 2223566100, 4941258000, 6617756250, 5672362500, 3151312500, 1102500000, 221484375, 19531250;\n","recurrence_sentence":"The row polynomials satisfy (7+5*n*x)*P(n,x) = (49+(35*n+70)*x+(25*n+25)*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 + (35*n + 70)*x + (25*n + 25)*x^2)*P[n - 1, x])/(7 + 5*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 + (35*n + 70)*x + (25*n + 25)*x^2)*v[i - 1])/(7 + 5*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-a6a633b989da1879","steps":["\u03b8+1"],"edges":["e-ff92d260041d3b3b"],"text":"A038271 \u2192 \u03b8+1"},"headline":"A038271 \u2192 \u03b8+1","chain":null}