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A228331 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(5).

%I A228331 #31 Feb 17 2025 01:33:07
%S A228331 1,36,780,16240,321300,6131664,114017904,2079380160,37356642180,
%T A228331 663144710800,11657925495216,203295462691776,3521108298744400,
%U A228331 60632838691387200,1038859802556120000,17721669103065158400,301147406355880764900,5099997408534884394000,86106549929771707182000
%N A228331 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(5).
%H A228331 Vincenzo Librandi, <a href="/A228331/b228331.txt">Table of n, a(n) for n = 0..200</a>
%H A228331 Pedro J. Miana and Natalia Romero, <a href="https://doi.org/10.1016/j.jnt.2010.01.018">Moments of combinatorial and Catalan numbers</a>, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega5. Remark 3 p. 1882.
%H A228331 Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2017">Four transformations on the Catalan triangle</a>, arXiv preprint arXiv:1305.2017, 2013
%H A228331 Yidong Sun and Fei Ma, <a href="https://doi.org/10.37236/3701">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33
%F A228331 Conjecture: n^2*a(n) +4*(2*n^2-22*n+11)*a(n-1) +16*(-7*n^2-54*n+166)*a(n-2) -1088*(2*n-3)*(2*n-7)*a(n-3)=0. - _R. J. Mathar_, Sep 08 2013
%F A228331 Recurrence: n^2*(3*n^2 - 5*n + 1)*a(n) = 4*(2*n-3)*(2*n+1)*(3*n^2 + n - 1)*a(n-1). - _Vaclav Kotesovec_, Dec 08 2013
%F A228331 a(n) = binomial(2*n,n)^2 * (2*n+1)*(3*n^2+n-1)/(2*n-1). - _Vaclav Kotesovec_, Dec 08 2013
%F A228331 G.f.: ((28*x+3)*hypergeom([1/2, 5/2],[1],16*x)+20*x*(1-16*x)*hypergeom([3/2, 7/2],[2],16*x))/3. - _Mark van Hoeij_, Apr 12 2014
%t A228331 Table[Sum[(k+1)^5*(Binomial[2n+1, n-k]*2*(k+1)/(n+k+2))^2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 08 2013 *)
%Y A228331 Cf. A000108, A039598, A024492, A000894, A228329, A000515, A228330, A228332, A228333.
%K A228331 nonn
%O A228331 0,2
%A A228331 _N. J. A. Sloane_, Aug 26 2013