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%I #5 Feb 21 2024 16:37:36
%S 1,17,163,1229,8131,49637,287323,1602989,8710291,46423157,243875083,
%T 1267233149,6530064451,33433332677,170320976443,864288997709,
%U 4372382138611,22066261554197,111150518391403,559034856752669,2808319611460771,14094228176783717
%N 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.
%C 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).
%F 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]
%K nonn
%O 0,2
%A Vesselin Dimitrov (avding(AT)hotmail.com), Mar 18 2002
%E More terms from _Sean A. Irvine_, Feb 21 2024
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