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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":"b8c08f1477c92f5517602ac43cf3e1ab6e5274edab98bc61cdbd6b1f1a3530c2","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-fe801e460afa1e3b","role":"sequence","name":"basis_map_A013609 of Square the entries of Pascal's triangle.","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:     2,       8;\n  2:     6,      48,       36;\n  3:    20,     240,      432,      128;\n  4:    70,    1120,     3240,     2560,       400;\n  5:   252,    5040,    20160,    26880,     12000,      1152;\n  6:   924,   22176,   113400,   215040,    168000,     48384,     3136;\n  7:  3432,   96096,   598752,  1478400,   1680000,    870912,   175616,     8192;\n  8: 12870,  411840,  3027024,  9225216,  13860000,  10644480,  3951360,   589824,   20736;\n  9: 48620, 1750320, 14826240, 53813760, 100900800, 103783680, 57953280, 16220160, 1866240, 51200;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+(6*n^3-3*n^2)*x+2*n^4*x^2)*P(n,x) = (8*n^2-4*n+(24*n^3+24*n^2-26*n)*x+(8*n^4+56*n^3+4*n^2-44*n+4)*x^2+(12*n^4+24*n^3-4*n^2-16*n)*x^3)*P(n-1,x) - (4*n*x+(12*n^2+20*n-1)*x^2+(4*n^3+36*n^2+32*n-2)*x^3+(8*n^3+24*n^2+16*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^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*P[n - 1, x] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*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^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*v[i - 1] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*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-b13531ad79e7c629","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-4579edc5f7a4d216","e-605949dedf8ba141"],"text":"A008459 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A008459 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}