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.
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
Offset: 0
Examples
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)
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.
- Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
- Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
- Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
- 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.
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.
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.
Programs
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Magma
A106566:= func< n,k | n eq 0 select 1 else (k/n)*Binomial(2*n-k-1, n-k) >; [A106566(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 06 2021
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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
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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 *) (* The function RiordanArray is defined in A256893. *) RiordanArray[1&, #(1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *) -
PARI
{T(n, k) = if( k<=0 || k>n, n==0 && k==0, binomial(2*n - k, n) * k/(2*n - k))}; /* Michael Somos, Oct 01 2022 */ -
Sage
def A106566(n, k): return 1 if (n==0) else (k/n)*binomial(2*n-k-1, n-k) flatten([[A106566(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 06 2021
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..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..n-k} T(n+k,2*k+j) = A039599(n,k).
Sum_{j>=0} T(n,j)*binomial(j,k) = A039599(n,k).
Sum_{k=0..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..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
Sum_{k=0..n} T(n,k)*A000045(k) = A109262(n), A000045: Fibonacci numbers. - Philippe Deléham, Oct 28 2008
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - x*z*c(z)) where c(z) the g.f. of A000108. - Michael Somos, Oct 01 2022
Extensions
Formula corrected by Philippe Deléham, Oct 31 2008
Corrected by Philippe Deléham, Sep 17 2009
Corrected by Alois P. Heinz, Aug 02 2012
Comments