This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A226974 #36 Feb 23 2025 07:04:18
%S A226974 1,1,1,2,5,11,25,64,169,443,1181,3224,8897,24701,69161,195255,554577,
%T A226974 1583109,4541461,13086574,37856437,109892403,320034309,934774902,
%U A226974 2737689189,8037746691,23652564261,69749727716,206091735797,610061655665,1808962146529
%N A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*binomial(4*k,k)/(3*k+1).
%H A226974 Vincenzo Librandi, <a href="/A226974/b226974.txt">Table of n, a(n) for n = 0..200</a>
%F A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*A002293(k).
%F A226974 Representation in terms of special values of hypergeometric function of type 6F5: a(n) = hypergeom([1/4, 1/2, 3/4, -(1/3)*n, -(1/3)*n+2/3, -(1/3)*n+1/3], [1/3, 2/3, 2/3, 1, 4/3],-4^4/3^3), n>=0.
%F A226974 Recurrence: 27*n*(n+1)*(n-1)*a(n) = 162*n*(n-1)^2*a(n-1) - 81*(5*n^2-15*n+12)*(n-1)*a(n-2) + 4*(199*n^3 - 1098*n^2 + 2043*n - 1296)*a(n-3) - (n-3)*(1173*n^2 - 5097*n + 5584)*a(n-4) + 6*(n-4)*(n-3)*(155*n-401)*a(n-5) - 283*(n-5)*(n-4)*(n-3)*a(n-6). - _Vaclav Kotesovec_, Jun 28 2013
%F A226974 a(n) ~ (3+4^(1+1/3))^(n+3/2)/(8*3^(n+1)*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 28 2013
%F A226974 G.f. satisfies A(x)=1+x^3*A(x)^4+x/(1-x). - _Vladimir Kruchinin_, May 17 2020
%F A226974 From _Peter Bala_, Sep 15 2021: (Start)
%F A226974 O.g.f.: A(x) = (1/x)*series reversion( x*(1 - x^3)/(1 + x*(1 - x^3)) ).
%F A226974 The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion( x*(1 - x^3)/(1 + (m + 1)*x*(1 - x^3)) ). The case m = -1 gives the sequence [1,0,0,1,0,0,4,0,0,22,0,0,140,...] - an aerated version of A002293. (End)
%p A226974 A226974 := proc(n)
%p A226974 hypergeom([-n/3,-n/3+2/3,-n/3+1/3,1/4,1/2,3/4],[1/3,2/3,2/3,1,4/3],-256/27) ;
%p A226974 simplify(%) ;
%p A226974 end proc:
%p A226974 seq(A226974(n),n=0..40) ; # _R. J. Mathar_, Jan 10 2023
%t A226974 Table[Sum[Binomial[n,3*k]*Binomial[4*k,k]/(3*k+1), {k,0,Floor[n/3]}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 28 2013 *)
%o A226974 (Maxima)
%o A226974 a(n):=if n<0 then 0 else if n=0 then 1 else sum(sum(sum(a(l)*a(i)*a(j)*a(n-i-j-l-3),l,0,n-3-i-j),j,0,n-3-i),i,0,n-3)+1; /* _Vladimir Kruchinin_, May 17 2020 */
%o A226974 (PARI) a(n) = sum(k=0, n\3, binomial(n,3*k)*binomial(4*k,k)/(3*k+1)); \\ _Michel Marcus_, Sep 16 2021
%Y A226974 Cf. A049130, A212385, A226910, A227035.
%K A226974 nonn,easy
%O A226974 0,4
%A A226974 _Karol A. Penson_, Jun 25 2013