OFFSET
0,3
COMMENTS
p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2).
p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).
Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1664 (terms 0..200 from Vincenzo Librandi)
FORMULA
a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!).
Recurrence: 2*n*(3*n-5)*a(n) = 3*(9*n^2 - 19*n + 8)*a(n-1) - 3*(n-1)*(3*n-4)*a(n-2) - 2*(2*n-3)*(3*n-2)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( (1 - x *c(x))/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k).
a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End)
MATHEMATICA
Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!), {i, 0, n}], {j, 0, n}], {n, 0, 50}]
PROG
(PARI) a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ Michel Marcus, Apr 02 2019
(PARI) a(n) = sum(i=0, 2*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^2, i)); \\ Seiichi Manyama, May 20 2019
(PARI) my(N=30, x='x+O('x^N)); Vec((1+sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x)))) \\ Seiichi Manyama, Jan 30 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Jul 14 2006
STATUS
approved