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%I #14 Sep 20 2017 04:28:28
%S 2,11,45,170,631,2346,8780,33089,125466,478181,1830258,7030557,
%T 27088856,104647615,405187809,1571990918,6109558567,23782190466,
%U 92705454875,361834392094,1413883873953,5530599237752,21654401079301,84859704298176
%N a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].
%C p divides a(p-1) and a(p-2) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
%C p divides a([(2p-1)/2]) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
%C p divides a((p-5)/2) for prime p=17,29,41,53,89,101.. =A040115[n] Primes of form 12n+5. Primes congruent to 5 (mod 12) excluding 5.
%C p divides a((p-5)/3) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.
%C p divides a([(p-3)/3]) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.
%H Vincenzo Librandi, <a href="/A120279/b120279.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}]. a(n) = A079309(n+1) - (n+1). a(n) = A066796(n+1)/2 - (n+1).
%F Recurrence: (n+1)*(3*n-2)*a(n) = 6*(3*n^2-1)*a(n-1) - 3*(9*n^2-n-2)*a(n-2) + 2*(2*n-1)*(3*n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 2^(2*n+3)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) = Sum_{k=1..n} Sum_{i=1..k} C(k+i,i). - _Wesley Ivan Hurt_, Sep 19 2017
%t Table[Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}],{n,1,50}]
%Y Cf. A007528, A007528, A040115, A048775, A079309, A079309, A066796.
%K nonn,easy
%O 1,1
%A _Alexander Adamchuk_, Jul 05 2006
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