A106534 Sum array of Catalan numbers (A000108) read by upward antidiagonals.

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 51, 36, 26, 19, 14, 188, 137, 101, 75, 56, 42, 731, 543, 406, 305, 230, 174, 132, 2950, 2219, 1676, 1270, 965, 735, 561, 429, 12235, 9285, 7066, 5390, 4120, 3155, 2420, 1859, 1430, 51822, 39587, 30302, 23236, 17846, 13726, 10571, 8151, 6292, 4862

Offset

0,2

Comments

The underlying array A is A(n, k) = Sum_{j=0..n} binomial(n, j)*A000108(k+j), n >= 0, k>= 0. See the example section. - Wolfdieter Lang, Oct 04 2019

Links

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

Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, page 5.

Formula

T(n, k) = 0 if k > n; T(n, n) = A000108(n); T(n, k) = T(n-1, k) + T(n, k+1) if 0 <= k < n.

T(n, k) = binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2], [k+2], -4). - Peter Luschny, Aug 16 2012

T(n, k) = A(n-k, k) = Sum_{j=0..n-k} binomial(n-k, j)*A000108(k+j), n >= 0, k = 0..n. - Wolfdieter Lang, Oct 03 2019

G.f.: (sqrt(1-4*x*y)-sqrt((5*x-1)/(x-1)))/(2*x*(x*y-y+1)). - Vladimir Kruchinin, Jan 12 2024

Example

From Wolfdieter Lang, Oct 04 2019: (Start)
The triangle T(n, k) begins:
n\k      0      1      2      3     4     5     6     7     8     9    10 ...
0:       1
1:       2      1
2:       5      3      2
3:      15     10      7      5
4:      51     36     26     19    14
5:     188    137    101     75    56    42
6:     731    543    406    305   230   174   132
7:    2950   2219   1676   1270   965   735   561   429
8:   12235   9285   7066   5390  4120  3155  2420  1859  1430
9:   51822  39587  30302  23236 17846 13726 10571  8151  6292  4862
10: 223191 171369 131782 101480 78244 60398 46672 36101 27950 21658 16796
... reformatted and extended.
-------------------------------------------------------------------------
The array A(n, k) begins:
n\k  0   1    2    3     4     5      6 ...
-------------------------------------------
0:   1   1    2    5    14    42    132 ... A000108
1    2   3    7   19    56   174    561 ... A005807
2:   5  10   26   75   230   735   2420 ...
3:  15  36  101  305   965  3155  10571 ...
4:  51 137  406 1270  4120 13726  46672 ...
5: 188 543 1676 5390 17846 60398 207963 ...
... (End)

Maple

# Uses floating point, precision might have to be adjusted.
C := n -> binomial(2*n, n)/(n+1);
H := (n, k) -> hypergeom([k-n, k+1/2], [k+2], -4);
T := (n, k) -> C(k)*H(n, k);
seq(print(seq(round(evalf(T(n, k), 32)), k=0..n)), n=0..7);
# Peter Luschny, Aug 16 2012

Mathematica

T[n_, n_] := CatalanNumber[n]; T[n_, k_] /; 0 <= k < n := T[n-1, k] + T[n, k+1]; T[_, _] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

Prog

(Sage)
def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) + T(n, k+1)
A106534 = lambda n, k: T(n, k)
for n in (0..5): [A106534(n, k) for k in (0..n)] # Peter Luschny, Aug 16 2012
(Magma)
function T(n, k)
  if k gt n then return 0;
  elif k eq n then return Catalan(n);
  else return T(n-1, k) + T(n, k+1);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 18 2021

Crossrefs

Columns: A007317, A002212, see also A045868, A055452-A055455.

Diagonals: A000108, A005807.

Cf. A059346 (Catalan difference array as triangle).

Sequence in context: A067418 A287548 A067323 * A123346 A163840 A122833

Adjacent sequences: A106531 A106532 A106533 * A106535 A106536 A106537

Keyword

nonn,easy,tabl

Author

Philippe Deléham, May 30 2005

Status

approved