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A143388
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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).
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1
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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; internal format)
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OFFSET
| 0,2
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FORMULA
| a(n) = (n^2 + 3*n + 6)*(3*n + 1)!/(n!*(2*n + 3)!) .
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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.
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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)!)}
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CROSSREFS
| Cf. A033184, A000108.
Sequence in context: A119817 A025570 A113449 * A027282 A006195 A092807
Adjacent sequences: A143385 A143386 A143387 * A143389 A143390 A143391
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 11 2008
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