A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.

1, 3, 26, 305, 4120, 60398, 934064, 15000903, 247766620, 4182015080, 71816825856, 1250772245698, 22039796891026, 392213323252200, 7038863826811100, 127248841020380105, 2315130641074743540, 42358284517663463380, 778876539384226875800

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 11.

Formula

a(n) = Sum_{i=0..n} binomial(n,i) * A000108(n+i).

D-finite with recurrence 2*n*(2*n+1)*(9*n-11)*a(n) +(-711*n^3+1589*n^2-986*n+144)*a(n-1) -10*(n-1)*(9*n-2)*(2*n-3)*a(n-2)=0.

Maple

A000108 := n-> binomial(2*n, n)/(n+1):
A283298 := proc(n)
    add(binomial(n, i)*A000108(n+i), i=0..n) ;
end proc:
seq(A283298(n), n=0..30) ;

Mathematica

Table[Sum[Binomial[n, i] CatalanNumber[n + i], {i, 0, n}], {n, 0, 50}] (* Indranil Ghosh, Jul 20 2017 *)

Prog

(Python)
from sympy import binomial, catalan
def a(n): return sum(binomial(n, i)*catalan(n + i) for i in range(n + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017
(PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = sum(i=0, n, binomial(n, i) * C(n+i)); \\ Michel Marcus, Nov 12 2022

Crossrefs

Central elements of rows in A106534, A280470.

Cf. A000108.

Sequence in context: A053972 A204561 A126738 * A259610 A326396 A109074

Adjacent sequences: A283295 A283296 A283297 * A283299 A283300 A283301

Keyword

nonn,easy

Author

R. J. Mathar, Jul 20 2017

Status

approved