A092472 - OEIS

%I #14 May 11 2013 01:56:20

%S 1,6,54,510,4830,45486,425502,3956238,36594558,337038702,3093092574,

%T 28302208974,258331692606,2353101799470,21397006320030,

%U 194281959853710,1761880227283710,15961196057303790,144466419007648350

%N a(n)=sum(i+j+k=n,(2n)!/(i+j)!/(j+k)!/(k+i)!) 0<=i<=n, 0<=j<=n, 0<=k<=n.

%H Vincenzo Librandi, <a href="/A092472/b092472.txt">Table of n, a(n) for n = 0..200</a>

%F Recurrence (for n>3): (n-3)*n*a(n) = (17*n^2-55*n+24)*a(n-1) - 36*(n-2)*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012

%F a(n) ~ 9^n. - _Vaclav Kotesovec_, Oct 14 2012

%F a(n) = 3^(2*n-1) * (2 + Sum_{k=1..n-2} 2^(k+2)*C(2*k+1,k-1)/3^(2*k+2) ), for n>2. - _Vaclav Kotesovec_, Oct 28 2012

%t Table[Sum[Sum[Sum[If[i+j+k==n,(2n)!/(i+j)!/(j+k)!/(k+i)!,0],{i,0,n}],{j,0,n}],{k,0,n}],{n,0,20}]

%t (* or *)

%t Flatten[{1,6,RecurrenceTable[{(n-3)*n*a[n]==(17*n^2-55*n+24)*a[n-1]-36*(n-2)*(2*n-3)*a[n-2],a[2]==54,a[3]==510},a,{n,2,20}]}] (* _Vaclav Kotesovec_, Oct 14 2012 *)

%t Flatten[{1,6,Table[3^(2*n-1)*(2+Sum[2^(k+2)*Binomial[2*k+1,k-1]/3^(2*k+2),{k,1,n-2}]),{n,2,20}]}] (* _Vaclav Kotesovec_, Oct 28 2012 *)

%o (PARI) a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,n,if(i+j+k-n,0,(2*n)!/(i+j)!/(j+k)!/(k+i)!))))

%K nonn

%O 0,2

%A _Benoit Cloitre_, Mar 25 2004