

A143388


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


1



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

0,2


REFERENCES

Ping Sun, Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes, Discrete Mathematics, 2017, in press; https://doi.org/10.1016/j.disc.2017.10.005


LINKS

Table of n, a(n) for n=0..20.


FORMULA

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


EXAMPLE

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,nk):
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.


PROG

(PARI) {a(n)=sum(k=0, n, binomial(2*nk, nk)*(k+1)/(n+1)*binomial(n+k, k)*(nk+1)/(n+1))}
(PARI) {a(n)=(n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!)}


CROSSREFS

Cf. A033184, A000108.
Sequence in context: A227081 A113449 A234938 * A027282 A006195 A214763
Adjacent sequences: A143385 A143386 A143387 * A143389 A143390 A143391


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Aug 11 2008


STATUS

approved



