%I #20 Feb 18 2019 04:07:26

%S 3,20,175,2076,32208,626028,14688924,404166432,12756813408,

%T 454171720320,18000130993920,785833199683200,37465916611881600,

%U 1936722997186387200,107888448215162361600,6442944965583133593600,410602312459198056960000

%N a(n) = n! * Sum_{k=1..n} binomial(2n+1,k)/k.

%C a(n) also equals (1/2)*Sum{k=1..n} (k+n)!*((k-1)!*4^k/(2*k)! + 1/(k! * k)).

%H G. C. Greubel, <a href="/A129840/b129840.txt">Table of n, a(n) for n = 1..365</a>

%F Recurrence: n*(2*n + 1)*(216*n^3 - 1062*n^2 + 1191*n - 131)*a(n) = (4320*n^6 - 25560*n^5 + 44808*n^4 - 19756*n^3 - 17153*n^2 + 18363*n - 4380)*a(n-1) - 3*(n-1)*(4752*n^6 - 34380*n^5 + 84072*n^4 - 68855*n^3 - 28120*n^2 + 65537*n - 21356)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*(4320*n^5 - 28800*n^4 + 57102*n^3 - 14749*n^2 - 49592*n + 22788)*a(n-3) - 8*(n-3)^2*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(216*n^3 - 414*n^2 - 285*n + 214)*a(n-4). - _Vaclav Kotesovec_, Mar 02 2014

%F a(n) ~ sqrt(Pi) * 2^(2*n+1/2) * n^(n-1/2) / exp(n) * (1 + 1/sqrt(n*Pi)). - _Vaclav Kotesovec_, Mar 02 2014

%p a:=n->n!*sum(binomial(2*n+1,k)/k,k=1..n): seq(a(n),n=1..20); # _Emeric Deutsch_, May 27 2007

%p A129840 := proc(n) n!*add(binomial(2*n+1,k)/k,k=1..n) ; end: for n from 1 to 20 do printf("%d, ",A129840(n)) ; od ; # _R. J. Mathar_, Jun 08 2007

%t Table[n!*Sum[Binomial[2n + 1, k]/k, {k, 1, n}], {n, 1, 25}] (* _Stefan Steinerberger_, May 24 2007 *)

%o (PARI) for(n=1,25, print1(n!*sum(k=1,n, binomial(2*n+1,k)/k), ", ")) \\ _G. C. Greubel_, Mar 20 2017

%K nonn

%O 1,1

%A _Leroy Quet_, May 22 2007

%E More terms from _R. J. Mathar_, _Stefan Steinerberger_ and _Emeric Deutsch_, May 24 2007