A110490 Diagonal sums of a triangle based on the Catalan numbers.

1, 1, 3, 7, 20, 59, 185, 600, 2003, 6833, 23727, 83606, 298313, 1076155, 3920823, 14416987, 53482012, 200151737, 755894009, 2882782933, 11115015138, 43400057683, 172016505877, 694208585423, 2863726993748, 12130698802645

Offset: 0

Author

Paul Barry, Jul 22 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of A110488.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..675

Crossrefs

Cf. A110488.

Programs

Mathematica

T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Join[{1, 1}, Table[Sum[T[n - k, k], {k, 0, n}], {n,2,50}]] (* G. C. Greubel, Aug 29 2017 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-2*k)} 2*(j+1)*(k-1)^j*C(2*(n-2*k)+1, n-2*k-j)/(n-2*k+j+2).