A068518 The sequence S(n,-3,1,1), where S(n,k,t,q) is defined by Sum_{j=0..n} binomial(n+q,j)^t * B(j,k) and B(j,k) is the j-th k-poly-Bernoulli number.

1, 17, 163, 1229, 8131, 49637, 287323, 1602989, 8710291, 46423157, 243875083, 1267233149, 6530064451, 33433332677, 170320976443, 864288997709, 4372382138611, 22066261554197, 111150518391403, 559034856752669, 2808319611460771, 14094228176783717

Offset: 0

Author

Vesselin Dimitrov (avding(AT)hotmail.com), Mar 18 2002

Keywords

• nonn

Comments

The sequence S(n,-2,1,1), n>=0, is A016269. It would be interesting to study the more general sequences S(r,-m,1,1), r=0,1,2,... for fixed m; here we consider the special cases m=3 and m=2. Finally, one can use the sum S(r,k,t,q) to discover certain recurrence relations involving poly-Bernoulli numbers. Let us note that the well known recurrence of the classical Bernoulli numbers yields S(r,1,1,1)=r+1. Let us also note that numerical experimentation suggests that S(r,-2,1,1)=S(r,-3,0,q).

Formula

a(n) = S(n, -3, 1, 1) = Sum_{k=0..n} ( binomial(n+1, k) * (-1)^k * Sum_{j=0..k} ((j+1)^3 * Sum_{i=0..j} (-1)^i * binomial(j, i) * i^k) ). [Corrected by Sean A. Irvine, Feb 21 2024]

Extensions

More terms from Sean A. Irvine, Feb 21 2024