A249454 E.g.f. exp(x*(sqrt(4*x^2+1)+2*x)).

1, 1, 5, 25, 121, 641, 4861, 44185, 272945, 480961, 19687861, 754778201, 1734823465, -259904463935, 180602875181, 188945735209561, 21689448762721, -167679491870531455, 2951642934531685, 193773398162963638681, 448885053392410841

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = (n-1)!*sum(k = 1..n, (4^(n-k)*binomial(n/2,n-k))/(k-1)!),n>0, a(0)=1.

a(n) ~ 2^((3*n-1)/2) * n^(n/2) / exp(n/2-1/4) if n is even, and a(n) ~ (-1)^((n+1)/2) * n^(n-1) * 2^n / exp(n+1/2) if n is odd. - Vaclav Kotesovec, Oct 31 2014

a(2*n+1) = 16^n*hypergeom([-2*n],[3/2-n],-1/4)*Gamma(3/2+n)/Gamma(3/2-n); a(2*n+1) ~ (-1)^(n+1)*2*(4*n/exp(1))^(2*n)/exp(1/2). - Peter Luschny, Oct 31 2014

D-finite with recurrence (-2*n+3)*a(n) +(-8*n^3+60*n^2-78*n-15)*a(n-2) +32*(n-2)*(n-3)*(n-4)*(2*n+1)*a(n-4)=0. - R. J. Mathar, Jul 27 2022

Maple

A249454 := proc(n)
    if n = 0 then
        1;
    else
        (n-1)!*add( 4^(n-k)*binomial(n/2, n-k)/(k-1)! , k=1..n) ;
    end if;
end proc;
seq(A249454(n), n=0..40) ; # R. J. Mathar, Jul 27 2022

Mathematica

CoefficientList[Series[E^(x*(Sqrt[4*x^2+1]+2*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 31 2014 *)

Prog

(Maxima)
a(n):=if n=0 then 1 else (n-1)!*sum((4^(n-k)*binomial(n/2, n-k))/(k-1)!, k, 1, n);

Crossrefs

Sequence in context: A218989 A078717 A275906 * A250347 A278282 A086093

Adjacent sequences: A249451 A249452 A249453 * A249455 A249456 A249457

Keyword

sign

Author

Vladimir Kruchinin, Oct 31 2014

Status

approved