A107111 Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 13, 22, 14, 1, 5, 23, 67, 90, 42, 1, 6, 36, 156, 381, 394, 132, 1, 7, 52, 305, 1162, 2307, 1806, 429, 1, 8, 71, 530, 2833, 9192, 14589, 8558, 1430, 1, 9, 93, 847, 5919, 27916, 75819, 95235, 41586, 4862, 1, 10, 118, 1272, 11070, 70098, 286632, 644908, 636925, 206098, 16796

Offset: 0

Author

Paul Barry, May 12 2005

Keywords

• easy
• nonn
• tabl

Comments

First row is the Catalan numbers A000108, second row is the large Schroeder numbers A006318, third row is A062992, fourth row is A007297. As a number triangle, this is T(n,k)=if(k<=n,sum{j=0..k, binomial((n-k)(k+1),k-j)*binomial(k+j,j)}/(k+1),0) with row sums A107112 and diagonal sums A107113.

Examples

			Array begins
1,1,2,5,14,42,132,...
1,2,6,22,90,394,1806,...
1,3,13,67,381,2307,14589,...
1,4,23,156,1162,9192,75819,...
		

Crossrefs

Cf. A366012.

Programs

Maple

A107111 := proc(n,k)
    add(binomial(n*(k+1),k-j)*binomial(k+j,j),j=0..k);
    %/(k+1) ;
end proc: # R. J. Mathar, Aug 02 2016

Mathematica

T[n_, k_] := Sum[Binomial[n (k + 1), k - j] Binomial[k + j, j], {j, 0, k}]/(k + 1);
Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

Formula

T(n, k)=sum{j=0..k, binomial(n(k+1), k-j)*binomial(k+j, j)}/(k+1)