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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
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a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(3*n-4*k+1,n-2*k).
1, 2, 8, 36, 181, 968, 5411, 31230, 184701, 1113534, 6818157, 42283904, 265051573, 1676628944, 10689175724, 68613428764, 443067507573, 2876254564034, 18759923273027, 122876716755094, 807909302669408, 5330342236103396, 35278723624832375
<a href="/index/Res#revert">Index entries for reversions of series</a>
allocated for Seiichi Manyama
Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x^2) ).
1, 2, 8, 36, 181, 968, 5411, 31230, 184701, 1113534, 6818157, 42283904, 265051573, 1676628944, 10689175724, 68613428764, 443067507573
0,2
(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^2+x^2))/x)
(PARI) a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(3*n-4*k+1, n-2*k))/(n+1);
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nonn
Seiichi Manyama, Jan 27 2024
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