This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 57
 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, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 Deléham, Nov 11 2007 LINKS Alois P. Heinz, Rows n = 0..140, flattened Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112. D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002. E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202. FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path, The number of initial rises of a Dyck paths, The number of nodes on the left branch of the tree, The number of subtrees. R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002. L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239. FORMULA 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 Deléham, Aug 25 2007 Sum_{k, 0<=k<=n}T(n,k)*A000108(k-1) = A121988(n), with A000108(-1)=0 . - Philippe Deléham, 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 Deléham, Oct 27 2007 T(n,k)*2^(n-k)=A110510(n,k) ; T(n,k)*3^(n-k)=A110518(n,k) . - Philippe Deléham, Nov 11 2007 Sum_{0<=k<=n}T(n,k)*A000045(k)=A109262(n), A000045: Fibonacci numbers. - Philippe Deléham, Oct 28 2008 Sum_{0<=k<=n}T(n,k)*A000129(k)=A143464(n), A000129: Pell numbers. - Philippe Deléham, Oct 28 2008 Sum_{0<=k<=n}T(n,k)*A100335(k)=A002450(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A100334(k)=A001906(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A099322(k)=A015565(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A106233(k)=A003462(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A151821(k+1)=A100320(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A082505(k+1)=A144706(n). - Philippe Deléham, Oct 30 2008 Sum_{0<=k<=n}T(n,k)*A000045(2k+2)=A026671(n). - Philippe Deléham, Feb 11 2009 Sum_{0<=k<=n}T(n,k)*A122367(k)=A026726(n). - Philippe Deléham, Feb 11 2009 Sum_{0<=k<=n}T(n,k)*A008619(k)=A000958(n+1). - Philippe Deléham, Nov 15 2009 Sum_{0<=k<=n}T(n,k)*A027941(k+1)=A026674(n+1).- Philippe Deléham, Feb 01 2014 EXAMPLE 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 From Paul Barry, 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) MAPLE A106566 := proc(n, k)     if n = 0 then         1;     elif k < 0 or k > n then         0;     else         binomial(2*n-k-1, n-k)*k/n ;     end if; end proc: # R. J. Mathar, Mar 01 2015 MATHEMATICA T[n_, k_] := Binomial[2n-k-1, n-k]*k/n; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2017 *) CROSSREFS 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, ... Sequence in context: A011434 A147746 A059365 * A099039 A205574 A049244 Adjacent sequences:  A106563 A106564 A106565 * A106567 A106568 A106569 KEYWORD nonn,tabl AUTHOR Philippe Deléham, May 30 2005 EXTENSIONS Corrected formula. - Philippe Deléham, Oct 31 2008 Corrected by Philippe Deléham, Sep 17 2009 Corrected by Alois P. Heinz, Aug 02 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.