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A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.

%I A381425 #109 Apr 11 2025 12:09:07
%S A381425 1,1,0,1,1,0,1,2,1,0,1,3,5,1,0,1,4,12,10,1,0,1,5,22,37,18,1,0,1,6,35,
%T A381425 92,102,30,1,0,1,7,51,185,345,258,47,1,0,1,8,70,326,880,1188,606,70,1,
%U A381425 0,1,9,92,525,1881,3851,3796,1335,100,1,0,1,10,117,792,3563,10002,15655,11364,2781,138,1,0
%N A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.
%H A381425 Seiichi Manyama, <a href="/A381425/b381425.txt">Antidiagonals n = 0..139, flattened</a>
%F A381425 A(n,k) = Sum_{j=0..k} binomial(k,j) * binomial(n+(k-1)*j-1,n-j).
%e A381425 Square array begins:
%e A381425   1, 1,  1,   1,    1,     1,     1, ...
%e A381425   0, 1,  2,   3,    4,     5,     6, ...
%e A381425   0, 1,  5,  12,   22,    35,    51, ...
%e A381425   0, 1, 10,  37,   92,   185,   326, ...
%e A381425   0, 1, 18, 102,  345,   880,  1881, ...
%e A381425   0, 1, 30, 258, 1188,  3851, 10002, ...
%e A381425   0, 1, 47, 606, 3796, 15655, 49468, ...
%o A381425 (PARI) a(n, k) = sum(j=0, k, binomial(k, j)*binomial(n+(k-1)*j-1, n-j));
%Y A381425 Columns k=0..2 give A000007, A000012, A177787.
%Y A381425 Rows n=0..3 give A000012, A001477, A000326, A096000(k-1).
%Y A381425 Main diagonal gives A382859.
%Y A381425 Cf. A071919, A362125.
%K A381425 nonn,easy,tabl
%O A381425 0,8
%A A381425 _Seiichi Manyama_, Apr 07 2025