0,8

Catalan convolution triangle; G.f. for column k : (x*c(x))^k with c(x) g.f. for A000108 (Catalan numbers).

Riordan array (1, xc(x)), where c(x) the g.f. of A000108; inverse of Riordan array (1, x(1-x)) [A109466] .

Diagonal sums give A132364 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2007

F. R. Bernhart, Catalan, Mozkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999), 167-202.

L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Paul Barry, A Catalan transform and related transformations on integer sequences, Journal of Integer Sequences, Vol. 8 (2005), pp. 1-24.

D. Callan, A recursive bijective approach to counting permutations . . .

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00. 1. 6

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations.

T(n, k) = binomial(2n-k-1, n-k)*k/n for 0<=k<=n with n>0; T(0, 0) = 1; T(0, k) = 0 if k>0.

T(0, 0) = 1; T(n, 0) = 0 if n>0; T(0, k) = 0 if k>0; for k>0 and n>0 : T(n, k) = Sum_{ j>=0 } T(n-1, k-1+j).

Sum_{ k>=0} T(n, k) = A000108(n).

Sum_{ j>=0} T(n+j, 2j) = binomial(2n-1, n), n>0.

Sum_{ j>=0} T(n+j, 2j+1) = binomial(2n-2, n-1), n>0.

Sum_{ k>=0} (-1)^(n+k)*T(n, k) = A064310(n). T(n, k) = (-1)^(n+k)*A099039(n, k).

Sum_{k>=0} T(n, k)*2^k = A000984(n).

Sum_{k>=0} T(n, k)*x^(n-k) = C(x, n); C(x, n) are the generalized Catalan numbers.

Sum_{k>=0} T(n, k)*3^k = A007854(n).

Sum_{k>=0} T(n, k)*4^k = A076035(n) .

Sum_{k>=0} T(n, k)*5^k = A076036(n).

Sum_{k, 0<=k<=n}T(n,k)*6^k = A127628(n) .

Sum_{j, 0<=j<=n-k}T(n+k,2*k+j)=A039599(n,k) .

Sum_{k, 0<=k<=n}T(n,k)*7^k = A126694(n) .

Sum_{j, j>=0}T(n,j)*binomial(j,k)=A039599(n,k).

Sum_{k, 0<=k<=n}T(n,k)*A000108(k)=A127632(n).

Sum_{k, 0<=k<=n}T(n,k)*(x+1)^k*x^(n-k)= A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x= 0,1,2,3,4,5,6,7,8,9 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007

Sum_{k, 0<=k<=n}T(n,k)*A000108(k-1) = A121988(n), with A000108(-1)=0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2007

Sum_{k, 0<=k<=n}T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 27 2007

T(n,k)*2^(n-k)=A110510(n,k) ; T(n,k)*3^(n-k)=A110518(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2007

Sum_{k, 0<=k<=n}T(n,k)*8^k = A115970(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2007

Triangle begins:

1

0 1

0 1 1

0 2 2 1

0 5 5 3 1

0 14 14 9 4 1

0 42 42 28 14 5 1

0 132 132 90 48 20 6 1

Column k for k = 0, 1, 2, ..., 13 : A000007, A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003517, A001392, A003518, A000589, A003519, A000590

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.

Diagonals : A000012, A001477, A000096, A005586, A005587, A005557, A064059, A064061

See also A009766, A033184, A059365 for other versions.

Generalized Catalan numbers C(x, n) for -11<=x<=10 : A064333, A064332, A064331, A064330, A064329, A064328, A064327, A064326, A064325, A064311, A064310, A000012, A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091, A064092, A064093.

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

Adjacent sequences: A106563 A106564 A106565 this_sequence A106567 A106568 A106569

Sequence in context: A011434 A059365 A099039 this_sequence A049244 A110281 A120059

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 30 2005