A120279 a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

2, 11, 45, 170, 631, 2346, 8780, 33089, 125466, 478181, 1830258, 7030557, 27088856, 104647615, 405187809, 1571990918, 6109558567, 23782190466, 92705454875, 361834392094, 1413883873953, 5530599237752, 21654401079301, 84859704298176

Offset: 1

Author

Alexander Adamchuk, Jul 05 2006

Keywords

• nonn
• easy

Comments

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.

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.

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.

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.

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A007528, A007528, A040115, A048775, A079309, A079309, A066796.

Programs

Mathematica

Table[Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}],{n,1,50}]

Formula

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).

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

a(n) ~ 2^(2*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

a(n) = Sum_{k=1..n} Sum_{i=1..k} C(k+i,i). - Wesley Ivan Hurt, Sep 19 2017