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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":"1cc203bd74f28a8b1f08889239ff7b94aabdaeb09ff13dd9f23b32b07f760850","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-c23f36ab5968876b","role":"sequence","name":"Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).","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,   2;\n  2: 1,  10,     3;\n  3: 1,  28,    63,       4;\n  4: 1,  60,   441,     340,        5;\n  5: 1, 110,  1881,    5632,     1705,         6;\n  6: 1, 182,  6006,   45760,    65065,      8190,         7;\n  7: 1, 280, 15834,  245960,   980265,    710892,     38227,        8;\n  8: 1, 408, 36414, 1005992,  8666515,  19531980,   7502593,   174760,      9;\n  9: 1, 570, 75582, 3385040, 53936155, 279527430, 372099553, 77344288, 786429, 10;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = -(n^2+2*n+1+(2*n^4-4*n^2+2)*x+(n^6-2*n^4+n^2)*x^2)*P(n-2,x) + ((2*n^3-6*n-4)*x^2+(2*n^5+n^4-4*n^3-3*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n+1)*x^3+(n^4+2*n^3+n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+2*n+1+2*n^4*x)*P(n-1,x) - (2*n^3-2*n^2)*x^2*d/dx P(n-1,x) + n^2*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 2*n + 1 + (2*n^4 - 4*n^2 + 2)*x + (n^6 - 2*n^4 + n^2)*x^2)*P[n - 2, x] + ((2*n^3 - 6*n - 4)*x^2 + (2*n^5 + n^4 - 4*n^3 - 3*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n + 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 2*n + 1 + 2*n^4*x)*P[n - 1, x] - (2*n^3 - 2*n^2)*x^2*D[P[n - 1, x], x] + n^2*x^3*D[P[n - 1, x], {x, 2}])/(n^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 2*n + 1 + (2*n^4 - 4*n^2 + 2)*x + (n^6 - 2*n^4 + n^2)*x^2)*v[i - 2] + ((2*n^3 - 6*n - 4)*x^2 + (2*n^5 + n^4 - 4*n^3 - 3*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n + 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 2*n + 1 + 2*n^4*x)*v[i - 1] - (2*n^3 - 2*n^2)*x^2*deriv(v[i - 1], x) + n^2*x^3*deriv(deriv(v[i - 1], x), x))/(n^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-c23f36ab5968876b","steps":["\u03b8+1"],"edges":["e-712d8024c16e1799"],"text":"A008957 \u2192 \u03b8+1"},"headline":"A008957 \u2192 \u03b8+1","chain":null}