This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336573 #45 Dec 22 2024 02:31:28
%S A336573 1,1,1,1,1,2,1,1,3,4,1,1,4,11,8,1,1,5,21,45,16,1,1,6,34,126,197,32,1,
%T A336573 1,7,50,267,818,903,64,1,1,8,69,484,2279,5594,4279,128,1,1,9,91,793,
%U A336573 5105,20540,39693,20793,256,1,1,10,116,1210,9946,56928,192350,289510,103049,512
%N A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
%C A336573 T(n,k) is the number of Sylvester classes of k-packed words of degree n.
%H A336573 Seiichi Manyama, <a href="/A336573/b336573.txt">Antidiagonals n = 0..139, flattened</a>
%H A336573 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.
%F A336573 G.f. A_k(x) of column k satisfies A_k(x) = 1 - x * A_k(x)^k * (1 - 2 * A_k(x)).
%F A336573 T(n,k) = ( (-1)^n / (k*n+1) ) * Sum_{j=0..n} (-2)^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
%F A336573 T(n,k) = (-1)^n*binomial(k*n+1, n)*hypergeom([-n, k*n+1], [(k-1)*n+2], 2)/(k*n+1) for k >= 1. - _Peter Luschny_, Jul 26 2020
%F A336573 T(n,k) = (1/n) * Sum_{j=0..n-1} binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - _Seiichi Manyama_, Aug 08 2023
%e A336573 Square array begins:
%e A336573 1, 1, 1, 1, 1, 1, ...
%e A336573 1, 1, 1, 1, 1, 1, ...
%e A336573 2, 3, 4, 5, 6, 7, ...
%e A336573 4, 11, 21, 34, 50, 69, ...
%e A336573 8, 45, 126, 267, 484, 793, ...
%e A336573 16, 197, 818, 2279, 5105, 9946, ...
%p A336573 T := (n,k) -> `if`(k=0, `if`(n=0, 1, 2^(n-1)), (-1)^n*(binomial(k*n+1, n)* hypergeom([-n, k*n+1], [(k-1)*n+2], 2)) / (k*n+1)):
%p A336573 seq(lprint(seq(simplify(T(n, k)), k=0..9)), n=0..6); # _Peter Luschny_, Jul 26 2020
%t A336573 T[n_, k_] := (-1)^n * Sum[(-2)^j * 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 A336573 (PARI) T(n, k) = (-1)^n*sum(j=0, n, (-2)^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
%o A336573 (PARI) T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^k*(1-2*A)); polcoeff(A, n);
%o A336573 (PARI) T(n, k) = (-1)^n*sum(j=0, n, (-2)^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);
%Y A336573 Columns k = 0-5 are: A011782, A001003, A003168, A243659, A243667, A243668.
%Y A336573 Main diagonal is A336495.
%Y A336573 Cf. A336534, A336574, A336575.
%K A336573 nonn,tabl
%O A336573 0,6
%A A336573 _Seiichi Manyama_, Jul 26 2020