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 A190907 Triangle read by rows: T(n,k) = binomial(n+k, n-k) k! / (floor(k/2)! * floor((k+2)/2)!). 2
 1, 1, 1, 1, 3, 1, 1, 6, 5, 3, 1, 10, 15, 21, 2, 1, 15, 35, 84, 18, 10, 1, 21, 70, 252, 90, 110, 5, 1, 28, 126, 630, 330, 660, 65, 35, 1, 36, 210, 1386, 990, 2860, 455, 525, 14, 1, 45, 330, 2772, 2574, 10010, 2275, 4200, 238, 126 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The triangle may be regarded as a generalization of the triangle A088617. A088617(n,k) = binomial(n+k,n-k)*(2*k)\$/(k+1); T(n,k) = binomial(n+k,n-k)*(k)\$ /(floor(k/2)+1). Here n\$ denotes the swinging factorial A056040(n). As A088617 is a decomposition of the large Schroeder numbers A006318, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected. T(n,n) = A057977(n) which can be seen as extended Catalan numbers. LINKS Peter Luschny, The lost Catalan numbers. FORMULA T(n,1) = A000217(n). T(n,2) = (n-1)*n*(n+1)*(n+2)/24 (Cf. A000332). EXAMPLE [0]  1 [1]  1,  1 [2]  1,  3,   1 [3]  1,  6,   5,   3 [4]  1, 10,  15,  21,   2 [5]  1, 15,  35,  84,  18,  10 [6]  1, 21,  70, 252,  90, 110,  5 [7]  1, 28, 126, 630, 330, 660, 65, 35 MAPLE A190907 := (n, k) -> binomial(n+k, n-k)*k!/(floor(k/2)!*floor((k+2)/2)!); seq(print(seq(A190907(n, k), k=0..n)), n=0..7); MATHEMATICA Flatten[Table[Binomial[n+k, n-k] k!/(Floor[k/2]!Floor[(k+2)/2]!), {n, 0, 10}, {k, 0, n}]] (* From Harvey P. Dale, May 05 2012 *) CROSSREFS Cf. Row sums: A190908; A056040, A085478, A088617, A060693. Sequence in context: A203950 A273349 A159572 * A035582 A156594 A109647 Adjacent sequences:  A190904 A190905 A190906 * A190908 A190909 A190910 KEYWORD nonn,tabl AUTHOR Peter Luschny, May 24 2011 STATUS approved

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Last modified October 17 10:07 EDT 2019. Contains 328108 sequences. (Running on oeis4.)