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%I #27 Jan 25 2024 10:47:25
%S 1,3,15,92,630,4620,35494,282015,2298417,19108265,161418543,
%T 1381606044,11955789440,104427062460,919430773992,8151530382264,
%U 72711166411422,652075100808960,5875868463764446,53175058170610530,483082193418731280,4404057834071995110
%N Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(4*n+2,n-3*k).
%F D-finite with recurrence 81*n*(n-1)*(n+1)*a(n) -945*n^2*(n-1)*a(n-1) +441*(n-1)*(3*n^2+9*n-20)*a(n-2) +3*(1039*n^3 -12393*n^2 +37406*n-33232)*a(n-3) -448*(2*n-5) *(4*n-13)*(4*n-11)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2024
%p A369114 := proc(n)
%p add(binomial(n+k,k) * binomial(4*n+2,n-3*k),k=0..floor(n/3)) ;
%p %/(n+1) ;
%p end proc;
%p seq(A369114(n),n=0..70) ; # _R. J. Mathar_, Jan 25 2024
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^3))/x)
%o (PARI) a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(4*n+2, n-3*k))/(n+1);
%Y Cf. A368011, A369102.
%Y Cf. A097188.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jan 13 2024
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