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_{ 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<=k<=n} T(n, k)*x^k = A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for x= 0,1,2,3,4,5,6,7,8 respectively .

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

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

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)*A000045(k)=A109262(n), A000045: Fibonacci numbers. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

Sum_{k, 0<=k<=n}T(n,k)*A000129(k)=A143464(n), A000129: Pell numbers. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

Sum_{k, 0<=k<=n}T(n,k)*A100335(k)=A002450(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A100334(k)=A001906(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A099322(k)=A015565(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A106233(k)=A003462(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A151821(k+1)=A100320(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A082505(k+1)=A144706(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A000045(2k+2)=A026671(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*A122367(k)=A026726(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

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

Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 28 2009: (Start)

Production array is

0, 1,

0, 1, 1,

0, 1, 1, 1,

0, 1, 1, 1, 1,

0, 1, 1, 1, 1, 1,

0, 1, 1, 1, 1, 1, 1,

0, 1, 1, 1, 1, 1, 1, 1,

0, 1, 1, 1, 1, 1, 1, 1, 1,

0, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)

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: A147746 A059365 A099039 this_sequence A049244 A110281 A120059

nonn,tabl

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

Corrected formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 31 2008

Corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009