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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 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.)