A228329 a(n) = Sum_{k=0..n} (k+1)^2*T(n,k)^2 where T(n,k) is the Catalan triangle A039598.

1, 8, 98, 1320, 18590, 268736, 3952228, 58837680, 883941750, 13373883600, 203487733020, 3110407163760, 47726453450988, 734694122886080, 11341161925265480, 175489379096245984, 2721169178975361702, 42273090191785999728, 657788911222324942060, 10250564041646388681200

Offset

0,2

Comments

Let h(m) denote the sequence whose n-th term is Sum_{k=0..n} (k+1)^m*T(n,k)^2, where T(n,k) is the Catalan triangle A039598. This is h(2).

Links

G. C. Greubel, Table of n, a(n) for n = 0..825

Pedro J. Miana, Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Remark 3 p. 1882. Omega2(n) = a(n-1).

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv:1305.2017 [math.CO], 2013.

Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Formula

Conjecture: n*(2*n+1)*a(n) + 2*(-26*n^2+25*n-11)*a(n-1) + 20*(4*n-5)*(4*n-7)*a(n-2) = 0. - R. J. Mathar, Sep 08 2013

a(n) = ((4n)!*(3n+1))/((2n)!^2*(2n+1)) = binomial(4n,2n)*(3n+1)/(2n+1). - Philippe Deléham, Nov 25 2013

From Peter Luschny, Nov 26 2013: (Start)

a(n) = 16^n*(3*n+1)*gamma(2*n+1/2)/(sqrt(Pi)*gamma(2*n+2)).

a(n) = a(n-1)*(6*n+2)*(4*n-3)*(4*n-1)/(n*(2*n+1)*(3*n-2)) if n > 0 else 1.

a(n) = [x^n] I*HeunG(8/5,0,-1/4,1/4,3/2,1/2,16*x)/sqrt(16*x-1) where [x^n] f(x) is the coefficient of x^n in f(x) and HeunG is the Heun general function. (End)

Maple

B:=(n, k)->binomial(2*n, n-k) - binomial(2*n, n-k-2); #A039598
Omega:=(m, n)->add((k+1)^m*B(n, k)^2, k=0..n);
h:=m->[seq(Omega(m, n), n=0..20)];
h(2);
# Second solution:
h := n -> I*HeunG(8/5, 0, -1/4, 1/4, 3/2, 1/2, 16*x)/sqrt(16*x-1);
seq(coeff(series(h(x), x, n+2), x, n), n=0..19); # Peter Luschny, Nov 26 2013

Mathematica

a[n_] := Binomial[4n, 2n] (3n+1)/(2n+1);
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 30 2018, after Philippe Deléham *)

Prog

(Sage)
@CachedFunction
def A228329(n):
    return A228329(n-1)*(6*n+2)*(4*n-3)*(4*n-1)/(n*(2*n+1)*(3*n-2)) if n>0 else 1
[A228329(n) for n in (0..19)]  # Peter Luschny, Nov 26 2013

Crossrefs

Cf. A039598, A000108, A024492 (h(0)), A000894 (h(1)), A000515 (h(3)), A228330 (h(4)), A228331 (h(5)) - A228333 (h(7)).

Cf. A000142, A007318.

Sequence in context: A116229 A099150 A199029 * A228794 A366035 A211869

Adjacent sequences: A228326 A228327 A228328 * A228330 A228331 A228332

Keyword

nonn

Author

N. J. A. Sloane, Aug 26 2013

Status

approved