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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":"3d993ccd33b46ca0de77806275e851e60bd6f00ff80ae6be40d39234bd8e007a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-107a00b591d3c4c8","role":"sequence","name":"basis_map_A013609 of Triangular array formed by the Mersenne numbers.","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,      4;\n  2:    14,     32,      12;\n  3:    48,    176,     144,      32;\n  4:   164,    832,    1080,     512,      80;\n  5:   560,   3632,    6528,    4864,    1600,     192;\n  6:  1912,  15104,   34848,   35840,   18400,    4608,     448;\n  7:  6528,  60800,  171648,  226560,  160000,   62208,   12544,   1024;\n  8: 22288, 239104,  799296, 1292288, 1169600,  623616,  194432,  32768,  2304;\n  9: 76096, 923968, 3571200, 6847488, 7589120, 5177088, 2207744, 573440, 82944, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(4*n+2*n*x-n^3*x^2)*P(n,x) = -(8*n+8+(2*n^2+12*n+10)*x-(2*n^3+2*n^2-4*n-4)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (4+12*x-(2*n^2+2*n-10)*x^2-(4*n^2+4*n-2)*x^3-(n^2+n)*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[(-(8*n + 8 + (2*n^2 + 12*n + 10)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (4 + 12*x - (2*n^2 + 2*n - 10)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-4*n - 2*n*x + n^3*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] = (-(8*n + 8 + (2*n^2 + 12*n + 10)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (4 + 12*x - (2*n^2 + 2*n - 10)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-4*n - 2*n*x + n^3*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-9b5c7e97e5191da6","steps":["basis A013609","\u03b8+1"],"edges":["e-3133312db52cd60e","e-576dea6e635e5cb8"],"text":"A110441 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A110441 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}