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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). 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 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 May 29 04:03 EDT 2023. Contains 363029 sequences. (Running on oeis4.)