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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":"7f38c9984159aabd236715d49954daeb0e1e5749702834103d4c7ddcee97ff0a","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-688fdf685007c036","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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:            289,            1360,              900;\n  1:           4913,           34680,            45900,             16000;\n  2:          83521,          786080,          1560600,           1088000,            250000;\n  3:        1419857,        16704200,         44217000,          46240000,          21250000,           3600000;\n  4:       24137569,       340765680,       1127533500,        1572160000,        1083750000,         367200000,          49000000;\n  5:      410338673,      6758519320,      26835297300,       46771760000,       42988750000,       21848400000,        5831000000,         640000000;\n  6:     6975757441,    131308375360,     608266738800,     1272191872000,     1461617500000,      990460800000,      396508000000,       87040000000,       8100000000;\n  7:   118587876497,   2511272678760,   13294973005200,    32440892736000,    44725495500000,    37885125600000,    20221908000000,     6658560000000,    1239300000000,     100000000000;\n  8:  2015993900449,  47435150598800,  282518176360500,   787850252160000,  1267222372500000,  1288094270400000,   859431090000000,   377318400000000,  105340500000000,   17000000000000,   1210000000000;\n  9: 34271896307633, 887037316197560, 5870099886601500, 18415999644240000, 33852940522500000, 40145604760800000, 32142722766000000, 17639635200000000, 6566224500000000, 1589500000000000, 226270000000000, 14400000000000;\n","recurrence_sentence":"The row polynomials satisfy (289+(510*n+850)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (4913+(8670*n+26010)*x+(1700*n^2+15300*n+28900)*x^2+(1000*n^2+6000*n+9000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1360*x + 900*x^2.","mathematica":"P[0, x_] := 289 + 1360*x + 900*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4913 + (8670*n + 26010)*x + (1700*n^2 + 15300*n + 28900)*x^2 + (1000*n^2 + 6000*n + 9000)*x^3)*P[n - 1, x])/(289 + (510*n + 850)*x + (100*n^2 + 400*n + 400)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1360*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (8670*n + 26010)*x + (1700*n^2 + 15300*n + 28900)*x^2 + (1000*n^2 + 6000*n + 9000)*x^3)*v[i - 1])/(289 + (510*n + 850)*x + (100*n^2 + 400*n + 400)*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-a5302200d285579d","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-2f341f062a4ee4be","e-bf2e839aa345a2d4"],"text":"A038331 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038331 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}