A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 14, 0, 1, 5, 14, 28, 42, 42, 0, 1, 6, 20, 48, 90, 132, 132, 0, 1, 7, 27, 75, 165, 297, 429, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 0

Offset

0,8

Comments

Reflected version of A106566.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.

Formula

T(n, k) = A106566(n, n-k).

Sum_{k=0..n} T(n,k) = A000108(n).

T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - Jean-François Alcover, Jun 14 2019

Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - G. C. Greubel, Jun 14 2022

G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.

Example

Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  2,   0;
  1, 3,  5,   5,   0;
  1, 4,  9,  14,  14,    0;
  1, 5, 14,  28,  42,   42,    0;
  1, 6, 20,  48,  90,  132,  132,    0;
  1, 7, 27,  75, 165,  297,  429,  429,    0;
  1, 8, 35, 110, 275,  572, 1001, 1430, 1430,    0;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862,  0;
  ...

Mathematica

T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 14 2019 *)

Prog

(Sage)
@CachedFunction
def A130020(n, k):
    if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
    return add(A130020(n-1, j) for j in (0..k))
for n in (0..10) :
    [A130020(n, k) for k in (0..n)]  # Peter Luschny, Nov 14 2012
(Magma)
A130020:= func< n, k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
[A130020(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2022
(PARI) {T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* Michael Somos, Oct 01 2022 */

Crossrefs

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

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

Cf. A000108 (Catalan numbers), A106566 (row reversal), A210736.

Sequence in context: A191646 A297321 A277938 * A292870 A306704 A091063

Adjacent sequences: A130017 A130018 A130019 * A130021 A130022 A130023

Keyword

nonn,tabl

Author

Philippe Deléham, Jun 16 2007

Status

approved