cp's OEIS Frontend

A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

%I A336534 #43 Aug 09 2025 20:53:49
%S A336534 1,1,2,1,2,2,1,2,6,2,1,2,10,22,2,1,2,14,66,90,2,1,2,18,134,498,394,2,
%T A336534 1,2,22,226,1482,4066,1806,2,1,2,26,342,3298,17818,34970,8558,2,1,2,
%U A336534 30,482,6202,52450,226214,312066,41586,2,1,2,34,646,10450,122762,881970,2984206,2862562,206098,2
%N A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
%H A336534 Seiichi Manyama, <a href="/A336534/b336534.txt">Antidiagonals n = 0..139, flattened</a>
%F A336534 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + A_k(x)).
%F A336534 T(n,k) = (1/n) * Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.
%F A336534 T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
%F A336534 T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -1)/(1 + k*n) for k > 0. - _Stefano Spezia_, Aug 09 2025
%e A336534 Square array begins:
%e A336534   1,   1,    1,     1,     1,      1, ...
%e A336534   2,   2,    2,     2,     2,      2, ...
%e A336534   2,   6,   10,    14,    18,     22, ...
%e A336534   2,  22,   66,   134,   226,    342, ...
%e A336534   2,  90,  498,  1482,  3298,   6202, ...
%e A336534   2, 394, 4066, 17818, 52450, 122762, ...
%t A336534 T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 01 2021 *)
%o A336534 (PARI) T(n, k) = sum(j=0, n, binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);
%Y A336534 Columns k=0-3 give A040000, A006318, A027307, A144097.
%Y A336534 If Michael D. Weiner's conjecture on A260332 is correct, column 4 is A260332 for n > 0.
%Y A336534 Main diagonal gives A336537.
%Y A336534 Cf. A336521, A336574. A336575.
%K A336534 nonn,tabl
%O A336534 0,3
%A A336534 _Seiichi Manyama_, Jul 25 2020