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

0,5

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.

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

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,...

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 *)

Crossrefs

Cf. A366012.

Sequence in context: A056860 A158825 A247507 * A082037 A163649 A110858

Adjacent sequences: A107108 A107109 A107110 * A107112 A107113 A107114

Keyword

easy,nonn,tabl

Author

Paul Barry, May 12 2005

Status

approved