%I A106566
%S A106566 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,
%T A106566 132,90,48,20,6,1,0,429,429,297,165,75,27,7,1,0,1430,1430,1001,
%U A106566 572,275,110,35,8,1,4862,4862,3432,2002,1001,429,154,44,9,1
%N A106566 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 
               1, . . . ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, . . . ] where DELTA is 
               the operator defined in A084938.
%C A106566 Catalan convolution triangle; G.f. for column k : (x*c(x))^k with c(x) 
               g.f. for A000108 (Catalan numbers).
%C A106566 Riordan array (1, xc(x)), where c(x) the g.f. of A000108; inverse of 
               Riordan array (1, x(1-x)) [A109466] .
%C A106566 Diagonal sums give A132364 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 11 2007
%D A106566 F. R. Bernhart, Catalan, Mozkin and Riordan numbers, Discr. Math., 204 
               (1999), 73-112.
%D A106566 E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999), 167-202.
%D A106566 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, 
               Discrete Applied Math., 34 (1991), 229-239.
%D A106566 Paul Barry, A Catalan transform and related transformations on integer 
               sequences, Journal of Integer Sequences, Vol. 8 (2005), pp. 1-24.
%H A106566 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive 
               bijective approach to counting permutations . . .</a>
%H A106566 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00. 1. 
               6</a>
%H A106566 A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/
               abs/math.CO/0203033">Refined restricted permutations</a>.
%F A106566 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.
%F A106566 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).
%F A106566 Sum_{ j>=0} T(n+j, 2j) = binomial(2n-1, n), n>0.
%F A106566 Sum_{ j>=0} T(n+j, 2j+1) = binomial(2n-2, n-1), n>0.
%F A106566 Sum_{ k>=0} (-1)^(n+k)*T(n, k) = A064310(n). T(n, k) = (-1)^(n+k)*A099039(n, 
               k).
%F A106566 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 .
%F A106566 Sum_{k>=0} T(n, k)*x^(n-k) = C(x, n); C(x, n) are the generalized Catalan 
               numbers.
%F A106566 Sum_{j, 0<=j<=n-k}T(n+k,2*k+j)=A039599(n,k) .
%F A106566 Sum_{j, j>=0}T(n,j)*binomial(j,k)=A039599(n,k).
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A000108(k)=A127632(n).
%F A106566 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
%F A106566 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
%F A106566 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
%F A106566 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
%F A106566 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]
%F A106566 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]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A100335(k)=A002450(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A100334(k)=A001906(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A099322(k)=A015565(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A106233(k)=A003462(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A151821(k+1)=A100320(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A082505(k+1)=A144706(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A000045(2k+2)=A026671(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Feb 11 2009]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A122367(k)=A026726(n). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Feb 11 2009]
%F A106566 Sum_{k, 0<=k<=n}T(n,k)*A008619(k)=A000958(n+1). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 15 2009]
%e A106566 Triangle begins:
%e A106566 1
%e A106566 0 1
%e A106566 0 1 1
%e A106566 0 2 2 1
%e A106566 0 5 5 3 1
%e A106566 0 14 14 9 4 1
%e A106566 0 42 42 28 14 5 1
%e A106566 0 132 132 90 48 20 6 1
%e A106566 Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 28 2009: (Start)
%e A106566 Production array is
%e A106566 0, 1,
%e A106566 0, 1, 1,
%e A106566 0, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1, 1, 1, 1, 1,
%e A106566 0, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
%Y A106566 Column k for k = 0, 1, 2, ..., 13 : A000007, A000108, A000108, A000245, 
               A002057, A000344, A003517, A000588, A003517, A001392, A003518, A000589, 
               A003519, A000590
%Y A106566 The three triangles A059365, A106566 and A099039 are the same except 
               for signs and the leading term.
%Y A106566 Diagonals : A000012, A001477, A000096, A005586, A005587, A005557, A064059, 
               A064061
%Y A106566 See also A009766, A033184, A059365 for other versions.
%Y A106566 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.
%Y A106566 The following are all versions of (essentially) the same Catalan triangle: 
               A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A106566 Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 
               A003519 A001392, ...
%Y A106566 Sequence in context: A147746 A059365 A099039 this_sequence A049244 A110281 
               A120059
%Y A106566 Adjacent sequences: A106563 A106564 A106565 this_sequence A106567 A106568 
               A106569
%K A106566 nonn,tabl,new
%O A106566 0,8
%A A106566 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 30 2005
%E A106566 Corrected formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 31 
               2008
%E A106566 Corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009