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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":"fa3d0fea7dcd8087c7f02d4f1552456f0f4226a8651dab614e1c931a071bd386","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-20086f90792ae22f","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:      5,       8;\n  2:     17,      48,      24;\n  3:     61,     248,     240,       64;\n  4:    217,    1152,    1656,      896,      160;\n  5:    773,    5064,    9648,     7872,     2880,     384;\n  6:   2753,   21456,   50904,    55552,    30880,    8448,     896;\n  7:   9805,   88600,  251568,   343872,   256960,  107136,   23296,    2048;\n  8:  34921,  358944, 1186680,  1949312,  1831200, 1029120,  341376,   61440,   4608;\n  9: 124373, 1432936, 5405808, 10371008, 11740480, 8313216, 3721984, 1021952, 156672, 10240;\n","recurrence_sentence":"The row polynomials satisfy (34*n-102+(17*n^2+95*n-234)*x+(4*n^3+54*n^2+70*n-180)*x^2+(8*n^3+44*n^2-4*n-48)*x^3+(8*n^2-8*n)*x^4)*P(n,x) = -(238*n+102+(187*n^2+733*n+234)*x+(28*n^3+338*n^2+690*n+180)*x^2+(24*n^3+156*n^2+188*n+48)*x^3+(8*n^2-8*n)*x^4)*P(n-1,x) + (289+(289*n+1377)*x+(68*n^2+1037*n+2546)*x^2+(148*n^2+1364*n+2302)*x^3+(96*n^2+792*n+1044)*x^4+(16*n^2+192*n+216)*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[(-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*P[n - 1, x] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*D[P[n - 1, x], x])/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*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] = (-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*v[i - 1] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*deriv(v[i - 1], x))/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*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-e646c30d067149e7","steps":["basis A007318","\u03b8+1"],"edges":["e-4395b7c136181822","e-2b9683ec965fecfe"],"text":"A249252 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A249252 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}