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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":"c8148f292d79e1d40c6c896b9bef5a8533d71c189c33f77134edd1ee9c80acd5","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-8d887bfc40d036b9","role":"sequence","name":"Cube of lower triangular normalized binomial matrix.","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:     3,      4;\n  2:     9,     24,      9;\n  3:    27,    108,     81,     16;\n  4:    81,    432,    486,    192,     25;\n  5:   243,   1620,   2430,   1440,    375,     36;\n  6:   729,   5832,  10935,   8640,   3375,    648,     49;\n  7:  2187,  20412,  45927,  45360,  23625,   6804,   1029,    64;\n  8:  6561,  69984, 183708, 217728, 141750,  54432,  12348,  1536,   81;\n  9: 19683, 236196, 708588, 979776, 765450, 367416, 111132, 20736, 2187, 100;\n","recurrence_sentence":"The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (27+(27*n+27)*x+(3*n^2+15*n+9)*x^2+(n^2+2*n+1)*x^3)*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[((27 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + n^2*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] = ((27 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + n^2*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-8d887bfc40d036b9","steps":["(\u03b8+1)\u00b2"],"edges":["e-32b8bc759b20a254"],"text":"A027465 \u2192 (\u03b8+1)\u00b2"},"headline":"A027465 \u2192 (\u03b8+1)\u00b2","chain":null}