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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":"da8e4ac430675b907f5f4bda5be0a9042adb6ce387aad931edd33e21f211dea5","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-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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:      2,       20,        27;\n  3:      6,      104,       315,        240;\n  4:     24,      616,      3060,       5040,        2625;\n  5:    120,     4176,     29736,      78400,       86625,       34020;\n  6:    720,    32112,    303660,    1128512,     1968750,     1621620,      509355;\n  7:   5040,   277056,   3310956,   16143680,    39289250,    50644440,    33108075,     8648640;\n  8:  40320,  2654784,  38719944,  236441920,   748940500,  1337295960,  1359638280,   735134400,   164189025;\n  9: 362880, 27999360, 485652024, 3594493056, 14169512500, 32742309600, 45877491660, 38399961600, 17677685025, 3445942500;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(4*n^2-4*n)*x-(9*n^2-16*n+7)*x^2)*P(n-2,x) - ((2*n-1)*x-(4*n^2-8*n)*x^2-(4*n^2+6*n-7)*x^3)*d/dx P(n-2,x) - (x^2-(4*n-4)*x^3-(4*n-3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+4*n*x)*P(n-1,x) + (2*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (4*n^2 - 4*n)*x - (9*n^2 - 16*n + 7)*x^2)*P[n - 2, x] - ((2*n - 1)*x - (4*n^2 - 8*n)*x^2 - (4*n^2 + 6*n - 7)*x^3)*D[P[n - 2, x], x] - (x^2 - (4*n - 4)*x^3 - (4*n - 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + 4*n*x)*P[n - 1, x] + (2*x - (4*n - 4)*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (4*n^2 - 4*n)*x - (9*n^2 - 16*n + 7)*x^2)*v[i - 2] - ((2*n - 1)*x - (4*n^2 - 8*n)*x^2 - (4*n^2 + 6*n - 7)*x^3)*deriv(v[i - 2], x) - (x^2 - (4*n - 4)*x^3 - (4*n - 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + 4*n*x)*v[i - 1] + (2*x - (4*n - 4)*x^2)*deriv(v[i - 1], 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-a36b2e24b7a3dbb7","steps":["(\u03b8+1)\u00b2"],"edges":["e-773358e866fad0c1"],"text":"A112486 \u2192 (\u03b8+1)\u00b2"},"headline":"A112486 \u2192 (\u03b8+1)\u00b2","chain":null}