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).

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, 1, 7, 50, 267, 818, 903, 64, 1, 1, 8, 69, 484, 2279, 5594, 4279, 128, 1, 1, 9, 91, 793, 5105, 20540, 39693, 20793, 256, 1, 1, 10, 116, 1210, 9946, 56928, 192350, 289510, 103049, 512

Offset

0,6

Comments

T(n,k) is the number of Sylvester classes of k-packed words of degree n.

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 - x * A_k(x)^k * (1 - 2 * A_k(x)).

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).

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

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

Example

Square array begins:
   1,   1,   1,    1,    1,    1, ...
   1,   1,   1,    1,    1,    1, ...
   2,   3,   4,    5,    6,    7, ...
   4,  11,  21,   34,   50,   69, ...
   8,  45, 126,  267,  484,  793, ...
  16, 197, 818, 2279, 5105, 9946, ...

Maple

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)):
seq(lprint(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, Jul 26 2020

Mathematica

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 *)

Prog

(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));
(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);
(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);

Crossrefs

Columns k = 0-5 are: A011782, A001003, A003168, A243659, A243667, A243668.

Main diagonal is A336495.

Cf. A336534, A336574, A336575.

Sequence in context: A256161 A137596 A111669 * A124834 A271465 A104495

Adjacent sequences: A336570 A336571 A336572 * A336574 A336575 A336576

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 26 2020

Status

approved