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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).

%I A143388 #15 May 31 2025 09:43:15
%S A143388 1,2,8,40,221,1288,7752,47652,297275,1874730,11920740,76292736,
%T A143388 490828828,3171317360,20563942288,133749903324,872196460359,
%U A143388 5700580759510,37332393806400,244914161562840,1609234420792845,10588423438256160,69757296470927520,460089876775105200
%N 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).
%H A143388 Ping Sun, <a href="https://doi.org/10.1016/j.disc.2017.10.005">Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes</a>, Discrete Mathematics, Volume 341, Issue 4, April 2018, Pages 1144-1149.
%F A143388 a(n) = (n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!).
%F A143388 a(n) ~ 3*sqrt(3)*(27/4)^n/(16*sqrt(n*Pi)). - _Stefano Spezia_, May 31 2025
%e A143388 Catalan triangle A033184 begins:
%e A143388    1;
%e A143388    1,  1;
%e A143388    2,  2,  1;
%e A143388    5,  5,  3,  1;
%e A143388   14, 14,  9,  4, 1;
%e A143388   42, 42, 28, 14, 5, 1;
%e A143388   ...
%e A143388 where column k equals the (k+1)-fold convolution of A000108, k>=0.
%e A143388 Illustrate a(n) = Sum_{k=0..n} A033184(n,k)*A033184(n,n-k):
%e A143388 a(1) = 1*1 + 1*1 = 2;
%e A143388 a(2) = 2*1 + 2*2 + 1*2 = 8;
%e A143388 a(3) = 5*1 + 5*3 + 3*5 + 1*5 = 40;
%e A143388 a(4) = 14*1 + 14*4 + 9*9 + 4*14 + 1*14 = 221.
%o A143388 (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))}
%o A143388 (PARI) {a(n)=(n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!)}
%Y A143388 Cf. A033184, A000108.
%K A143388 nonn,tabl
%O A143388 0,2
%A A143388 _Paul D. Hanna_, Aug 11 2008
%E A143388 a(21)-a(23) from _Stefano Spezia_, May 31 2025