cp's OEIS Frontend

A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.

%I A349640 #19 Mar 14 2025 08:35:31
%S A349640 1,2,7,46,485,7066,130987,2946182,77923561,2369742130,81467904431,
%T A349640 3124302688222,132237820201357,6123150708289226,307903794151741075,
%U A349640 16709463201832993846,973385368533058021457,60583668821975488285282,4012342371757905842648791,281735471040327667890013070
%N A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.
%C A349640 For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 2, 1, 0, 1, 2, 2, 1, 0, ...] with period 5. In particular, a(5*n+4) == 0 (mod 5). Cf. A047974. - _Peter Bala_, Mar 13 2025
%F A349640 a(n) ~ 2^(2*n + 1/2) * n^(n-1) / exp(n - 1/4).
%F A349640 From _Peter Luschny_, Nov 23 2021: (Start)
%F A349640 a(n) = n! * [x^n](exp(x)*(1 - sqrt(1 - 4*x))/(2*x)).
%F A349640 a(n) = (4*(n-1)*(n-2)*a(n - 3) - (n-1)*(8*n-5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1) for n >= 4.
%F A349640 a(n-1) = A224500(n) / n for n >= 1. (End)
%F A349640 a(n) = hypergeom([-n, 1/2, 1], [2], -4). - _Peter Bala_, Mar 13 2025
%p A349640 gf := exp(x)*(1 - sqrt(1 - 4*x))/(2*x): ser := series(gf, x, 24):
%p A349640 seq(n!*coeff(ser, x, n), n = 0..19);
%p A349640 # Alternative:
%p A349640 a := n -> `if`(n < 4, [1, 2, 7, 46][n + 1], ((4*n^2 - 12*n + 8)*a(n - 3) - (8*n^2 - 13*n + 5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1)):
%p A349640 seq(a(n), n = 0..19);  # _Peter Luschny_, Nov 23 2021
%p A349640 # Alternative
%p A349640 seq(simplify(hypergeom([-n, 1/2, 1], [2], -4)), n = 0..19); # _Peter Bala_, Mar 13 2025
%t A349640 Table[Sum[Binomial[n, j]*CatalanNumber[j]*j!, {j, 0, n}], {n, 0, 20}]
%o A349640 (PARI) a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k!); \\ _Michel Marcus_, Nov 23 2021
%Y A349640 Cf. A007317, A064613, A292632, A349603, A349639, A224500.
%K A349640 nonn
%O A349640 0,2
%A A349640 _Vaclav Kotesovec_, Nov 23 2021