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%I #11 Jan 28 2024 09:57:51
%S 1,2,8,36,181,968,5411,31230,184701,1113534,6818157,42283904,
%T 265051573,1676628944,10689175724,68613428764,443067507573,
%U 2876254564034,18759923273027,122876716755094,807909302669408,5330342236103396,35278723624832375
%N Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x^2) ).
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(3*n-4*k+1,n-2*k).
%F D-finite with recurrence -4*(101081336359*n -250960227225)*(2*n+1)*(n+2)*(n+1)*a(n) +2*(n+1)*(4508236721003*n^3 -9655124154789*n^2 -3820459908080*n +1505761363350)*a(n-1) +(-23789427607131*n^4 +67773978800606*n^3 +6302004268491*n^2 -64689912723806*n +2007681817800)*a(n-2) +6*(-890478123851*n^4 +42952117976042*n^3 -249768239921769*n^2 +474601169757458*n -268271866959440)*a(n-3) +4*(19581924322759*n^4 -271221111012910*n^3 +1384197210338720*n^2 -3056763018536945*n +2448325905713826)*a(n-4) -8*(n-4) *(15937841315391*n^3 -115485075434884*n^2 +337125583432496*n -379272346578549)*a(n-5) +16*(n-4)*(n-5) *(2696300795657*n^2 -3846744412865*n -4519001936313)*a(n-6) +2720*(105762416493*n -349473414130)*(n-4)*(n-5)*(n-6)*a(n-7)=0. - _R. J. Mathar_, Jan 28 2024
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^2+x^2))/x)
%o (PARI) a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(3*n-4*k+1, n-2*k))/(n+1);
%Y Cf. A369618, A369620.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jan 27 2024
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