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A336575 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 3^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.

%I A336575 #38 Aug 09 2025 20:51:56
%S A336575 1,1,3,1,3,3,1,3,12,3,1,3,21,57,3,1,3,30,192,300,3,1,3,39,408,2001,
%T A336575 1686,3,1,3,48,705,6402,22539,9912,3,1,3,57,1083,14799,109137,267276,
%U A336575 60213,3,1,3,66,1542,28488,338430,1964010,3287496,374988,3,1,3,75,2082,48765,817743,8181597,36718680,41556585,2381322,3
%N A336575 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 3^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.
%H A336575 Seiichi Manyama, <a href="/A336575/b336575.txt">Antidiagonals n = 0..139, flattened</a>
%F A336575 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (2 + A_k(x)).
%F A336575 T(n,k) = Sum_{j=0..n} 2^(n-j) * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
%F A336575 T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^j * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
%F A336575 T(n,k) = (1/n) * Sum_{j=0..n-1} (-2)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - _Seiichi Manyama_, Aug 10 2023
%F A336575 T(n,k) = 3*hypergeom([1-n, -k*n], [2], 3) for n > 0. - _Stefano Spezia_, Aug 09 2025
%e A336575 Square array begins:
%e A336575   1,    1,     1,      1,      1,      1, ...
%e A336575   3,    3,     3,      3,      3,      3, ...
%e A336575   3,   12,    21,     30,     39,     48, ...
%e A336575   3,   57,   192,    408,    705,   1083, ...
%e A336575   3,  300,  2001,   6402,  14799,  28488, ...
%e A336575   3, 1686, 22539, 109137, 338430, 817743, ...
%t A336575 T[0, k_] := 1; T[n_, k_] := Sum[3^j * Binomial[n, j] * Binomial[k*n, j - 1], {j, 1, n}]/n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 27 2020 *)
%o A336575 (PARI) T(n, k) = if(n==0, 1, sum(j=1, n, 3^j*binomial(n, j)*binomial(k*n, j-1))/n);
%o A336575 (PARI) T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(2+A)); polcoeff(A, n);
%o A336575 (PARI) T(n, k) = sum(j=0, n, 2^(n-j)*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
%o A336575 (PARI) T(n, k) = sum(j=0, n, 2^j*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);
%Y A336575 Columns k=0-4 give: A122553, A047891, A219535, A336538, A336540.
%Y A336575 Main diagonal gives A336578.
%Y A336575 Cf. A336534, A336573, A336574.
%K A336575 nonn,tabl
%O A336575 0,3
%A A336575 _Seiichi Manyama_, Jul 26 2020