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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

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Last modified December 10 20:48 EST 2017. Contains 295856 sequences.