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a(n) = Sum_{k=0..n} C(k*(k+1)/2, k) * C(k*(k+1)/2, n-k).
1

%I #9 Oct 05 2020 08:40:43

%S 1,1,4,29,339,5406,109159,2664399,76219485,2499425650,92402751894,

%T 3801481338219,172231146665554,8520038462375370,456913020454609665,

%U 26402792230144908683,1635399597258002744628,108090275300425856404653

%N a(n) = Sum_{k=0..n} C(k*(k+1)/2, k) * C(k*(k+1)/2, n-k).

%H Robert Israel, <a href="/A137646/b137646.txt">Table of n, a(n) for n = 0..364</a>

%F a(n) ~ c * d^n * (n-1)!, where d = 2/(LambertW(2*exp(-2))*(2 + LambertW(2*exp(-2)))) and c = 0.1589194832536846052272420789649724553661575944775731783119884812477... - _Vaclav Kotesovec_, Oct 05 2020

%p f:= proc(n) add(binomial(k*(k+1)/2,k)*binomial(k*(k+1)/2,n-k),k=ceil((sqrt(8*n+9)-3)/2)..n) end proc:

%p map(f, [$0..20]); # _Robert Israel_, Jun 07 2018

%t Join[{1},Table[Sum[Binomial[(k(k+1))/2,k]Binomial[(k(k+1))/2,n-k],{k,n}],{n,20}]] (* _Harvey P. Dale_, Jul 09 2018 *)

%o (PARI) a(n)=sum(k=0,n,binomial(k*(k+1)/2,k)*binomial(k*(k+1)/2,n-k))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 31 2008