A106534 - OEIS

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

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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);