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 A354242 Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)). 7
 1, 2, 14, 158, 2486, 50222, 1239254, 36126638, 1214933846, 46299580142, 1971815255894, 92809525295918, 4784166929982806, 268050260650705262, 16219498558371118934, 1054102762745609325998, 73229184033780135425366, 5415407651703010175897582 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Peter Bala, Jul 07 2022: (Start) Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 2, 14, 14, 6, 14, 6, 14, 6, ...], with an apparent period of 2 beginning at a(3). Cf. A354253. More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End) LINKS Seiichi Manyama, Table of n, a(n) for n = 0..360 FORMULA E.g.f.: Sum_{k>=0} binomial(2*k,k) * (exp(x) - 1)^k. a(n) = Sum_{k=0..n} (2*k)! * Stirling2(n,k)/k!. a(n) ~ sqrt(2/5) * n^n / (exp(n) * log(5/4)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022 Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x/(1 - 5*x/(1 - 6*x/(1 - 10*x/(1 - 10*x/(1 - 15*x/(1 - ... - (4*n-2)*x/(1 - 5*n*x/(1 - ...))))))))). - Peter Bala, Jul 07 2022 a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023 PROG (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(5-4*exp(x)))) (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(exp(x)-1)^k))) (PARI) a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 2)/k!); CROSSREFS Cf. A305404, A354252, A354253. Cf. A000670, A001813, A094417, A316747, A354240, A354241. Sequence in context: A196791 A349312 A218295 * A268011 A052112 A354241 Adjacent sequences: A354239 A354240 A354241 * A354243 A354244 A354245 KEYWORD nonn AUTHOR Seiichi Manyama, May 20 2022 STATUS approved

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Last modified September 24 22:14 EDT 2023. Contains 365582 sequences. (Running on oeis4.)