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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":"4e0ce44914b6119fb27be79b87c13383837e1ec2140d0c2dddd6fbbf18a02dda","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8652466ad45b52e6","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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:       1;\n  2:       2,        2;\n  3:       7,       14,        3;\n  4:      35,      102,       54,        4;\n  5:     226,      862,      738,      164,        5;\n  6:    1787,     8416,     9969,     3952,      440,       6;\n  7:   16717,    93648,   141795,    82240,    17925,    1098,      7;\n  8:  180560,  1172282,  2162232,  1660580,   559175,   73350,   2618,    8;\n  9: 2211181, 16322474, 35505354, 34032436, 15716825, 3357654, 280336, 6056, 9;\n","recurrence_sentence":"The row polynomials satisfy (x+(n-1)*x^3)*P(n,x) = (2*n-2-(n^2-5*n+7)*x+(4*n-4)*x^2-(n^3-4*n^2+4*n+2)*x^3-(2*n^3-6*n^2+4*n)*x^4-(n^3-3*n^2+2*n)*x^5)*P(n-2,x) + (1-(n-1)*x-(n^2-2*n+1)*x^3-(n^2-n+1)*x^4+(n^2-n)*x^5+(n^2-n)*x^6)*d/dx P(n-2,x) - (2-(2*n-1)*x+2*x^2-(2*n^2-2*n)*x^3-(2*n^2-4*n+2)*x^4)*P(n-1,x) + (x+(n-2)*x^3-(n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n^2 - 5*n + 7)*x + (4*n - 4)*x^2 - (n^3 - 4*n^2 + 4*n + 2)*x^3 - (2*n^3 - 6*n^2 + 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*P[n - 2, x] + (1 - (n - 1)*x - (n^2 - 2*n + 1)*x^3 - (n^2 - n + 1)*x^4 + (n^2 - n)*x^5 + (n^2 - n)*x^6)*D[P[n - 2, x], x] - (2 - (2*n - 1)*x + 2*x^2 - (2*n^2 - 2*n)*x^3 - (2*n^2 - 4*n + 2)*x^4)*P[n - 1, x] + (x + (n - 2)*x^3 - (n - 1)*x^5)*D[P[n - 1, x], x])/(x + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n^2 - 5*n + 7)*x + (4*n - 4)*x^2 - (n^3 - 4*n^2 + 4*n + 2)*x^3 - (2*n^3 - 6*n^2 + 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*v[i - 2] + (1 - (n - 1)*x - (n^2 - 2*n + 1)*x^3 - (n^2 - n + 1)*x^4 + (n^2 - n)*x^5 + (n^2 - n)*x^6)*deriv(v[i - 2], x) - (2 - (2*n - 1)*x + 2*x^2 - (2*n^2 - 2*n)*x^3 - (2*n^2 - 4*n + 2)*x^4)*v[i - 1] + (x + (n - 2)*x^3 - (n - 1)*x^5)*deriv(v[i - 1], x))/(x + (n - 1)*x^3)); 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-7b6a59413dc1b477","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-121b28b1325b14d7","e-39b8fa0cb0c06472"],"text":"A102365 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A102365 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}