0,4

The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the second to-left column is A001700, the alternating sum in each row but the first is 0.

T(n,k) is the number of Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g. T(n,k)=C(n,k)C(2n-k,n-1)/n for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003

A090181*A007318 as infinite lower triangular matrices . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 14 2008]

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.

G. E. Cossali, A Common Generating Function of Catalan Numbers and Other Integer Sequences, J. Int. Seqs. 6 (2003).

If C_n(x) is the gf of row n of the Narayana numbers (A001263), C_n(x) = sum({n choose k-1}{n-1 choose k-1}/k x^k, {k,1,n}) and T_n(x) is the gf of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]sum(A001263(n,k) (x+1)^k, (k,1,n)) - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007, Jan 31 2007

G.f.: (1-ty-sqrt((1-yt)^2-4y))/2.

T(n, k) = C(2n-k, n)*C(n, k)/(n-k+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 07 2003

A060693(n, k) = C(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 30 2003

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x=-1,0,1,2,3,4,5,6,7,8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 01 2007

T(n,k)=Sum_[j, j>=0}A090181(n,j)*binomial(j,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 04 2007

Sum_[k, 0<=k<=n}T(n,k)*x^(n-k)= (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x= -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 29 2009: (Start)

G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);

G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)

T(n,k)=[k<=n]*sum{j=0..n, C(n,j)^2*C(j,k)}/(n-k+1). [From Paul Barry (pbarry(AT)wit.ie), May 28 2009]

{1}, {1,1}, {2,3,1}, {5,10,6,1}, {14,35,30,10,1}, ...

Cf. A006318, A000108, A001700.

Triangle in A088617 transposed.

Diagonals give : A000108 A001700 A002457 A002802 A002803, A000012 A000217 A034827 A000910 A088625 A088626.

Sequence in context: A030103 A105640 A090299 this_sequence A089302 A049020 A144634

Adjacent sequences: A060690 A060691 A060692 this_sequence A060694 A060695 A060696

nonn,tabl

Frederic Chapoton (chapoton(AT)math.jussieu.fr), Apr 20 2001

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 21 2001

New description from Philippe Deleham, Aug 12, 2003