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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":"d7d78d0a311711069d2abd719067f3cdd3ecc94e6e3b087b9c176a4a2422f9ba","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-359eaa7b22014e19","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:     4,      6;\n  2:    13,     36,      18;\n  3:    43,    174,     171,      48;\n  4:   142,    756,    1107,     624,     120;\n  5:   469,   3096,    6021,    5100,    1980,     288;\n  6:  1549,  12204,   29592,   33456,   19590,    5760,     672;\n  7:  5116,  46842,  136053,  192408,  150765,   66960,   15792,   1536;\n  8: 16897, 176292,  596583, 1012944,  994725,  591984,  211008,  41472,   3456;\n  9: 55807, 653538, 2524941, 5003532, 5907810, 4411170, 2108232, 626304, 105408, 7680;\n","recurrence_sentence":"The row polynomials satisfy (26*n-78+(13*n^2+73*n-180)*x+(3*n^3+42*n^2+53*n-138)*x^2+(6*n^3+36*n^2-6*n-36)*x^3+(8*n^2-8*n)*x^4)*P(n,x) = -(130*n+78+(104*n^2+404*n+180)*x+(15*n^3+186*n^2+385*n+138)*x^2+(12*n^3+84*n^2+108*n+36)*x^3+(4*n^2-4*n)*x^4)*P(n-1,x) + (169+(169*n+819)*x+(39*n^2+624*n+1558)*x^2+(87*n^2+863*n+1480)*x^3+(60*n^2+548*n+732)*x^4+(12*n^2+156*n+176)*x^5+(16*n+16)*x^6)*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[(-(130*n + 78 + (104*n^2 + 404*n + 180)*x + (15*n^3 + 186*n^2 + 385*n + 138)*x^2 + (12*n^3 + 84*n^2 + 108*n + 36)*x^3 + (4*n^2 - 4*n)*x^4)*P[n - 1, x] + (169 + (169*n + 819)*x + (39*n^2 + 624*n + 1558)*x^2 + (87*n^2 + 863*n + 1480)*x^3 + (60*n^2 + 548*n + 732)*x^4 + (12*n^2 + 156*n + 176)*x^5 + (16*n + 16)*x^6)*D[P[n - 1, x], x])/(26*n - 78 + (13*n^2 + 73*n - 180)*x + (3*n^3 + 42*n^2 + 53*n - 138)*x^2 + (6*n^3 + 36*n^2 - 6*n - 36)*x^3 + (8*n^2 - 8*n)*x^4)]];\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] = (-(130*n + 78 + (104*n^2 + 404*n + 180)*x + (15*n^3 + 186*n^2 + 385*n + 138)*x^2 + (12*n^3 + 84*n^2 + 108*n + 36)*x^3 + (4*n^2 - 4*n)*x^4)*v[i - 1] + (169 + (169*n + 819)*x + (39*n^2 + 624*n + 1558)*x^2 + (87*n^2 + 863*n + 1480)*x^3 + (60*n^2 + 548*n + 732)*x^4 + (12*n^2 + 156*n + 176)*x^5 + (16*n + 16)*x^6)*deriv(v[i - 1], x))/(26*n - 78 + (13*n^2 + 73*n - 180)*x + (3*n^3 + 42*n^2 + 53*n - 138)*x^2 + (6*n^3 + 36*n^2 - 6*n - 36)*x^3 + (8*n^2 - 8*n)*x^4)); 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-e7331b0d50084c15","steps":["basis A007318","\u03b8+1"],"edges":["e-212843e2c5c5a487","e-9919baec3f451a6f"],"text":"A249251 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A249251 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}