generalization of A005251, A118891

[Max Alekseyev]

SeqFans,

As a side result of my search I came to the following paper:

http://www.emis.de/journals/MV/9434/mv943407.ps

that may serve as a source of new sequences in OEIS and a new
characterization of some existing ones.

In particular:

a_2(n) is A005251

a_3(n) is A118891 (or A049864 of which A118891 is a subsequence; btw,
I vote to merge A118891 with A049864 as A118891 is simply missing
several initial terms; a situation is similar to A005251 for which no
duplicate sequence starting with the 3rd term exists, instead it
contains a comment "a(n+3) = number ..." that is a way to go)

a_4(n) is missing in OEIS.
A straightforward PARI code and initial 0th to 20th terms:

{ a4(n) = 1 + sum(i=1,n, sum(j=0,n-i, sum(k=0,(n-i-j)\2,
sum(l=0,(n-i-j-2*k)\4,
binomial(i-1,j)*binomial(i-1-j,k)*binomial(i-1-j-2*k,l)*binomial(n-i-j-2*k-3*l+1,l+1)
)))) }
? vector(21,n,a2(n-1))
%1 = [1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897,
3329, 5842, 10252, 17991, 31572, 55405, 97229]

a_k(n) for k>4 are also missing in OEIS

Would anybody like to compute and add some of missing sequences (at
least a_4(n)) as well as a square array T(k,n) = a_k(n) (including all
aforementioned sequences) to OEIS ? It would also make sense to add a
link to the paper to existing sequences and interconnect these
sequences via %Y fields.

Unfortunately, I have no time for this now.

P.S. Please note that there is a typo in the paper in the formula for
a_2(n): the upper bound in the inner sum should be "n-i" not "i-1".

Regards,
Max