A143165 Expansion of the exponential generating function arcsin(2*x)/(2*(1-2*x)^(3/2)).

0, 1, 6, 49, 468, 5469, 73362, 1138005, 19737000, 383284665, 8163588510, 190709475705, 4818820261500, 131650382056725, 3850053335966250, 120466494638624925, 4002649276431128400, 141156781966460192625, 5252646220794868029750, 206149276075766825426625

Offset: 0

Author

Wolfdieter Lang, Sep 15 2008

Keywords

• nonn
• easy

Comments

Used in A024199(n+1) = A003148(n) + a(n).

Binomial convolution of [0,1^2,0,2^2,0,...,0,((2*k)!/k!)^2,0,...] (e.g.f. arcsin(2*x)/2) with the double factorials A001147.

Examples

			a(3) + A003148(3) = 49 + 27 = 76 = A024199(4).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..402

Crossrefs

Cf. A003148, A024199.

Programs

Maple

f:= gfun:-rectoproc({2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0, a(0)=0,a(1)=1,a(2)=6},a(n),remember):
map(f, [$0 .. 30]); # Robert Israel, Feb 07 2018

Mathematica

With[{nn=20},CoefficientList[Series[ArcSin[2x]/(2(1-2x)^(3/2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 18 2019 *)

PARI

x = 'x + O('x^40); concat(0, Vec(serlaplace(asin(2*x)/(2*(1-2*x)^(3/2))))) \\ Michel Marcus, Jun 18 2017

Formula

E.g.f.: arcsin(2*x)/(2*(1-2*x)^(3/2)).

a(n) = sum(binomial(n,2*k+1)*(4^k)*((2*k-1)!!)^2*(2*(n-2*k)-1)!!,k=0..floor(n/2)), with (2*n-1)!!:= A001147(n) (double factorials).

a(n) ~ Pi * 2^(n-1/2) * n^(n+1) / exp(n) * (1 - sqrt(2/(Pi*n))). - Vaclav Kotesovec, Mar 18 2014

2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0. - Robert Israel, Feb 07 2018