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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). 1
1, 2, 8, 40, 221, 1288, 7752, 47652, 297275, 1874730, 11920740, 76292736, 490828828, 3171317360, 20563942288, 133749903324, 872196460359, 5700580759510, 37332393806400, 244914161562840, 1609234420792845 (list; table; graph; refs; listen; history; text; internal format)
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,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.

PROG

(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)!)}

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

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Last modified May 13 02:36 EDT 2021. Contains 343836 sequences. (Running on oeis4.)