login
This site is supported by donations to The OEIS Foundation.

 

Logo


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

%I

%S 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 132,90,48,20,6,1,0,429,429,297,165,75,27,7,1,0,1430,1430,1001,572,

%U 275,110,35,8,1,0,4862,4862,3432,2002,1001,429,154,44,9,1

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

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

%C Diagonal sums give A132364. - _Philippe Deléham_, Nov 11 2007

%H Alois P. Heinz, <a href="/A106566/b106566.txt">Rows n = 0..140, flattened</a>

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.

%H D. Callan, <a href="http://arxiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations containing 3-letter patterns</a>, arXiv:math/0211380 [math.CO], 2002.

%H E. Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00371-9">Dyck path enumeration</a>, Discrete Math., 204, 1999, 167-202.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000011">The number of touch points of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000025">The number of initial rises of a Dyck paths</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000061">The number of nodes on the left branch of the tree</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000084">The number of subtrees</a>.

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

%H A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.

%H L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan group</a>, Discrete Applied Math., 34 (1991), 229-239.

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

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

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

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

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

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

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

%F 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

%F Sum_{k, 0<=k<=n}T(n,k)*A000108(k-1) = A121988(n), with A000108(-1)=0 . - _Philippe Deléham_, Aug 27 2007

%F 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

%F T(n,k)*2^(n-k)=A110510(n,k) ; T(n,k)*3^(n-k)=A110518(n,k) . - _Philippe Deléham_, Nov 11 2007

%F Sum_{0<=k<=n}T(n,k)*A000045(k)=A109262(n), A000045: Fibonacci numbers. - _Philippe Deléham_, Oct 28 2008

%F Sum_{0<=k<=n}T(n,k)*A000129(k)=A143464(n), A000129: Pell numbers. - _Philippe Deléham_, Oct 28 2008

%F Sum_{0<=k<=n}T(n,k)*A100335(k)=A002450(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A100334(k)=A001906(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A099322(k)=A015565(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A106233(k)=A003462(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A151821(k+1)=A100320(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A082505(k+1)=A144706(n). - _Philippe Deléham_, Oct 30 2008

%F Sum_{0<=k<=n}T(n,k)*A000045(2k+2)=A026671(n). - _Philippe Deléham_, Feb 11 2009

%F Sum_{0<=k<=n}T(n,k)*A122367(k)=A026726(n). - _Philippe Deléham_, Feb 11 2009

%F Sum_{0<=k<=n}T(n,k)*A008619(k)=A000958(n+1). - _Philippe Deléham_, Nov 15 2009

%F Sum_{0<=k<=n}T(n,k)*A027941(k+1)=A026674(n+1).- _Philippe Deléham_, Feb 01 2014

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 2 2 1

%e 0 5 5 3 1

%e 0 14 14 9 4 1

%e 0 42 42 28 14 5 1

%e 0 132 132 90 48 20 6 1

%e From _Paul Barry_, Sep 28 2009: (Start)

%e Production array is

%e 0, 1,

%e 0, 1, 1,

%e 0, 1, 1, 1,

%e 0, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)

%p A106566 := proc(n,k)

%p if n = 0 then

%p 1;

%p elif k < 0 or k > n then

%p 0;

%p else

%p binomial(2*n-k-1,n-k)*k/n ;

%p end if;

%p end proc: # _R. J. Mathar_, Mar 01 2015

%t 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 *)

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

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

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

%Y See also A009766, A033184, A059365 for other versions.

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

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

%K nonn,tabl

%O 0,8

%A _Philippe Deléham_, May 30 2005

%E Corrected formula. - _Philippe Deléham_, Oct 31 2008

%E Corrected by _Philippe Deléham_, Sep 17 2009

%E Corrected by _Alois P. Heinz_, Aug 02 2012

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.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 22 05:58 EST 2018. Contains 299430 sequences. (Running on oeis4.)