0,2

Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.

T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC}, and {AD,BD}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004

P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 16 2003

Essentially same triangle as triangle T(n, k), n>0 and k>0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deleham's operator defined in AO84938.

T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0}T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k<0 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004

T(n, k)=(k+1)sum(binomial(n+1, k+j+1)*binomial(n+j, j), j=0..n-k)/(n+1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004

Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

Rows are {1}, {2,1}, {6, 4, 1}, {22, 16, 6, 1}, ...

T:=(n, k)->(k+1)*sum(binomial(n+1, k+j+1)*binomial(n+j, j), j=0..n-k)/(n+1): seq(seq(T(n, k), k=0..n), n=0..9);

Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0, 0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1, k-1] + w[n-1, k] + w[n, k+1] Table[w[n, k], {n, 0, 10}, {k, 0, n}] - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

Cf. A000007 A033877 A084938.

Sequence in context: A110681 A117852 A080245 this_sequence A078937 A132159 A112356

Adjacent sequences: A080244 A080245 A080246 this_sequence A080248 A080249 A080250

nonn,tabl

Paul Barry (pbarry(AT)wit.ie), Feb 15 2003