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A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

%I A369617 #21 Mar 04 2024 14:06:00
%S A369617 1,4,22,146,1079,8525,70468,601816,5268241,47019566,426250277,
%T A369617 3914020148,36328457669,340278596273,3212416054283,30534649412247,
%U A369617 291981031204917,2806832429353512,27109863184695640,262951127248539898,2560229132085602215
%N A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).
%H A369617 Seiichi Manyama, <a href="/A369617/b369617.txt">Table of n, a(n) for n = 0..984</a>
%H A369617 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369617 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*n-4*k+2,n-k).
%F A369617 D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 28 2024
%p A369617 A369617 := proc(n)
%p A369617     add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
%p A369617     %/(n+1) ;
%p A369617 end proc;
%p A369617 seq(A369617(n),n=0..70) ; # _R. J. Mathar_, Jan 28 2024
%o A369617 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^3+x))/x)
%o A369617 (PARI) a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-4*k+2, n-k))/(n+1);
%Y A369617 Cf. A007317, A369616, A370844.
%Y A369617 Cf. A006632, A274735.
%K A369617 nonn
%O A369617 0,2
%A A369617 _Seiichi Manyama_, Jan 27 2024