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%I #29 Jan 30 2024 08:17:01
%S 1,1,5,49,697,12881,291901,7823425,241878449,8469678817,331194361141,
%T 14301627569681,675802760007145,34681947121134769,1920727213363900397,
%U 114166002761833118881,7248797582463164166241,489621781318487529974465
%N Expansion of e.g.f. exp(x/(1-4*x)^(1/2)).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="https://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F E.g.f.: exp(x/(1-4*x)^(1/2)).
%F a(n) = n!*sum((sum(2^k*k/(n-m)*binomial(2*(n-m)-k-1,n-m-1)*binomial(k+m-1,m-1),k,1,n-m))/m!,m,1,n-1)+1. - _Vladimir Kruchinin_, Sep 10 2010
%F Recurrence (for n>5): (n-5)*a(n) = 6*(2*n^2 - 13*n + 16)*a(n-1) - (48*n^3 - 432*n^2 + 1199*n - 1051)*a(n-2) + 2*(n-2)*(4*n-15)*(8*n^2 - 54*n + 89)*a(n-3) + 4*(n-4)*(n-3)*(n-2)*a(n-4). - _Vaclav Kotesovec_, Jun 27 2013
%F a(n) ~ n^(n-1/3)*exp(3*n^(1/3)/4-n)*4^n/sqrt(6). - _Vaclav Kotesovec_, Jun 27 2013
%F a(n) = n! * Sum_{k=0..n} 4^(n-k) * binomial(n-k/2-1,n-k)/k!. - _Seiichi Manyama_, Jan 30 2024
%t CoefficientList[Series[E^(x/(1-4*x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o (Maxima) a(n):=n!*sum((sum(2^k*k/(n-m)*binomial(2*(n-m)-k-1,n-m-1)*binomial(k+m-1,m-1),k,1,n-m))/m!,m,1,n-1)+1; /* _Vladimir Kruchinin_, Sep 10 2010 */
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-4*x)))) \\ _Joerg Arndt_, Jan 30 2024
%Y Cf. A362158.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 23 2000
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