A120305 - OEIS

%I #40 Apr 07 2024 17:43:24

%S 1,1,3,9,31,111,407,1513,5679,21471,81643,311895,1196131,4602235,

%T 17757183,68680169,266200111,1033703055,4020716123,15662273839,

%U 61092127491,238582873475,932758045123,3650336341239,14298633670931

%N a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

%C 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).

%C p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).

%C Essentially the same as partial sums of A072547. - _Seiichi Manyama_, Jan 30 2023

%H Seiichi Manyama, <a href="/A120305/b120305.txt">Table of n, a(n) for n = 0..1664</a> (terms 0..200 from Vincenzo Librandi)

%F a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!).

%F 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

%F a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 13 2013

%F 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

%F From _Seiichi Manyama_, Apr 06 2024: (Start)

%F a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k).

%F a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End)

%t Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!),{i,0,n}],{j,0,n}],{n,0,50}]

%o (PARI) a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ _Michel Marcus_, Apr 02 2019

%o (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

%o (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

%Y Cf. A000108, A006134, A033200, A007519, A092785, A307354.

%Y Cf. A072547, A371798.

%K nonn

%O 0,3

%A _Alexander Adamchuk_, Jul 14 2006