A143388 a(n) = Sum_{k=0..n} A033184(n,k)*A033184(n,n-k), where Catalan triangle entry A033184(n,k) = C(2*n-k,n-k)*(k+1)/(n+1).

1, 2, 8, 40, 221, 1288, 7752, 47652, 297275, 1874730, 11920740, 76292736, 490828828, 3171317360, 20563942288, 133749903324, 872196460359, 5700580759510, 37332393806400, 244914161562840, 1609234420792845, 10588423438256160, 69757296470927520, 460089876775105200

Offset: 0

Author

Paul D. Hanna, Aug 11 2008

Keywords

• nonn
• tabl

Examples

			Catalan triangle A033184 begins:
   1;
   1,  1;
   2,  2,  1;
   5,  5,  3,  1;
  14, 14,  9,  4, 1;
  42, 42, 28, 14, 5, 1;
  ...
where column k equals the (k+1)-fold convolution of A000108, k>=0.
Illustrate a(n) = Sum_{k=0..n} A033184(n,k)*A033184(n,n-k):
a(1) = 1*1 + 1*1 = 2;
a(2) = 2*1 + 2*2 + 1*2 = 8;
a(3) = 5*1 + 5*3 + 3*5 + 1*5 = 40;
a(4) = 14*1 + 14*4 + 9*9 + 4*14 + 1*14 = 221.
		

Links

• Ping Sun, Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes, Discrete Mathematics, Volume 341, Issue 4, April 2018, Pages 1144-1149.

Crossrefs

Cf. A033184, A000108.

Programs

PARI

{a(n)=sum(k=0,n,binomial(2*n-k,n-k)*(k+1)/(n+1)*binomial(n+k,k)*(n-k+1)/(n+1))}

PARI

{a(n)=(n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!)}

Formula

a(n) = (n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!).

a(n) ~ 3*sqrt(3)*(27/4)^n/(16*sqrt(n*Pi)). - Stefano Spezia, May 31 2025

Extensions

a(21)-a(23) from Stefano Spezia, May 31 2025