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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":"c00eac626f788bed4e58532495988ab60629411aad5d438bf4f526e13f81ceb0","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-cf9f646a76b9c283","role":"sequence","name":"A triangular array related to A077028 and distributing the values of A007582.","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,    4;\n  2: 1,   10,    12;\n  3: 1,   20,    51,    32;\n  4: 1,   38,   153,   196,     80;\n  5: 1,   72,   402,   784,    645,    192;\n  6: 1,  138,   990,  2600,   3225,   1926,    448;\n  7: 1,  268,  2349,  7760,  12875,  11556,   5383,   1024;\n  8: 1,  526,  5439, 21644,  44975,  53886,  37681,  14344,  2304;\n  9: 1, 1040, 12372, 57568, 143710, 215376, 200900, 114752, 36873, 5120;\n","recurrence_sentence":"The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2-(2*n-12)*x-(3*n^2-11*n)*x^2-(n^3-3*n^2-2*n+4)*x^3)*P(n-1,x) + (2*x+3*n*x^2+(n^2+3*n-4)*x^3+(n^2-2)*x^4)*d/dx 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[((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*P[n - 1, x] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*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] = ((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*v[i - 1] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*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-cf9f646a76b9c283","steps":["\u03b8+1"],"edges":["e-7e2621111628e31c"],"text":"A110552 \u2192 \u03b8+1"},"headline":"A110552 \u2192 \u03b8+1","chain":null}