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A228329 a(n) = Sum_{k=0..n} (k+1)^2*T(n,k)^2 where T(n,k) is the Catalan triangle A039598. 8
1, 8, 98, 1320, 18590, 268736, 3952228, 58837680, 883941750, 13373883600, 203487733020, 3110407163760, 47726453450988, 734694122886080, 11341161925265480, 175489379096245984, 2721169178975361702, 42273090191785999728, 657788911222324942060, 10250564041646388681200 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

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

LINKS

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

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

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: A099150 A116229 A199029 * A228794 A211869 A159232

Adjacent sequences:  A228326 A228327 A228328 * A228330 A228331 A228332

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 26 2013

STATUS

approved

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Last modified October 24 00:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)