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 A194689 a(n) = Sum_{k=0..n} binomial(n,k)*w(k)*w(n-k) where w() = A000296(). 10
 1, 0, 2, 2, 14, 42, 222, 1066, 6078, 36490, 238046, 1653610, 12214270, 95361866, 784071966, 6764984362, 61066919230, 575200190986, 5640081557598, 57450510336234, 606773139773054, 6633515763375306, 74950634205257630, 873995513192234410, 10504736507220958142, 129983468625156713354 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 771, Problem 37). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..560 FORMULA G.f.: 1/Q(0) where Q(k) = 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013 G.f.: 1/Q(0), where Q(k)= 1 - x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013 E.g.f.: exp(2*(exp(x) - 1 - x)). - Ilya Gutkovskiy, Apr 07 2018 a(0) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-1-k). - Seiichi Manyama, Nov 20 2020 a(n) ~ 4 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). - Vaclav Kotesovec, Jun 26 2022 MATHEMATICA Table[Sum[(-1)^(n-k) * Binomial[n, k] * BellB[k, 2] * 2^(n-k), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 25 2022 *) PROG (PARI) N=66; x='x+O('x^N); Q(k) = if (k>N, 1, 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ) ); gf=1/Q(0); Vec(Ser(gf)) /* Joerg Arndt, Mar 07 2013 */ (PARI) my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1-x)))) \\ Seiichi Manyama, Nov 20 2020 CROSSREFS Cf. A000296, A001861, A194689, A339014, A339017, A339027. Sequence in context: A235349 A226157 A264508 * A277556 A321179 A345370 Adjacent sequences: A194686 A194687 A194688 * A194690 A194691 A194692 KEYWORD nonn AUTHOR N. J. A. Sloane, Sep 01 2011 STATUS approved

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Last modified June 9 01:44 EDT 2023. Contains 363168 sequences. (Running on oeis4.)