cp's OEIS Frontend

A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.

%I A362078 #11 Apr 08 2023 11:07:03
%S A362078 1,1,1,1,1,3,1,1,5,10,1,1,7,22,35,1,1,9,37,105,126,1,1,11,55,215,511,
%T A362078 462,1,1,13,76,369,1271,2534,1716,1,1,15,100,571,2526,7651,12720,6435,
%U A362078 1,1,17,127,825,4401,17577,46614,64449,24310,1,1,19,157,1135,7026,34412,123810,286599,328900,92378
%N A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.
%F A362078 T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-n,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(n+j-1,j) * binomial(k*j,n-j).
%e A362078 Square array begins:
%e A362078     1,   1,    1,    1,    1,    1, ...
%e A362078     1,   1,    1,    1,    1,    1, ...
%e A362078     3,   5,    7,    9,   11,   13, ...
%e A362078    10,  22,   37,   55,   76,  100, ...
%e A362078    35, 105,  215,  369,  571,  825, ...
%e A362078   126, 511, 1271, 2526, 4401, 7026, ...
%o A362078 (PARI) T(n, k) = sum(j=0, n, binomial(n+j-1, j)*binomial(k*j, n-j));
%Y A362078 Columns k=0..3 give A088218, A213684, A362087, A362088.
%Y A362078 Main diagonal gives A362080.
%Y A362078 Cf. A362079.
%K A362078 nonn,tabl
%O A362078 0,6
%A A362078 _Seiichi Manyama_, Apr 08 2023