A052142 Expansion of e.g.f. exp(x/(1-4*x)^(1/2)).

1, 1, 5, 49, 697, 12881, 291901, 7823425, 241878449, 8469678817, 331194361141, 14301627569681, 675802760007145, 34681947121134769, 1920727213363900397, 114166002761833118881, 7248797582463164166241, 489621781318487529974465

Offset: 0

Author

N. J. A. Sloane, Jan 23 2000

Keywords

• nonn

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.

Links

• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A362158.

Programs

Mathematica

CoefficientList[Series[E^(x/(1-4*x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

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 */

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-4*x)))) \\ Joerg Arndt, Jan 30 2024

Formula

E.g.f.: exp(x/(1-4*x)^(1/2)).

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

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

a(n) ~ n^(n-1/3)*exp(3*n^(1/3)/4-n)*4^n/sqrt(6). - Vaclav Kotesovec, Jun 27 2013

a(n) = n! * Sum_{k=0..n} 4^(n-k) * binomial(n-k/2-1,n-k)/k!. - Seiichi Manyama, Jan 30 2024