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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":"f10a241decfbf7ec2f8b53e5cd18a7fb1b8c5ae5ca79c1fceeb2830cea12998f","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-10f70e3ea8b205cf","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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:         11,           8;\n  2:        121,         176,          36;\n  3:       1331,        2904,        1188,         128;\n  4:      14641,       42592,       26136,        5632,        400;\n  5:     161051,      585640,      479160,      154880,      22000,       1152;\n  6:    1771561,     7730448,     7906140,     3407360,     726000,      76032,      3136;\n  7:   19487171,    99207416,   121754556,    65591680,   18634000,    2927232,    241472,     8192;\n  8:  214358881,  1247178944,  1785733488,  1154413568,  409948000,   85865472,  10624768,   720896,   20736;\n  9: 2357947691, 15433839432, 25255373616, 19047823872, 8116970400, 2125170432, 350617344, 35684352, 2052864, 51200;\n","recurrence_sentence":"The row polynomials satisfy (121+(66*n-22)*x+4*n^2*x^2)*P(n,x) = (1331+(726*n+726)*x+(44*n^2+220*n+132)*x^2+(8*n^2+16*n+8)*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[((1331 + (726*n + 726)*x + (44*n^2 + 220*n + 132)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(121 + (66*n - 22)*x + 4*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] = ((1331 + (726*n + 726)*x + (44*n^2 + 220*n + 132)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(121 + (66*n - 22)*x + 4*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-a920b56c9b3ef4d1","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-71aa91d8d70f875b","e-6fd47421d0f3241e"],"text":"A038303 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038303 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}