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%I #30 Jan 25 2024 10:40:35
%S 1,0,2,2,13,28,127,376,1522,5210,20403,74952,292313,1114704,4371839,
%T 17040586,67378981,266402370,1061919289,4241539218,17030430061,
%U 68554148388,276988107861,1121954081852,4557637048543,18556386241468,75729621399950
%N Expansion of (1/x) * Series_Reversion( x * (1-x^2/(1-x))^2 ).
%H Seiichi Manyama, <a href="/A368957/b368957.txt">Table of n, a(n) for n = 0..1000</a>
%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(2*n+k+1,k) * binomial(n-k-1,n-2*k).
%F D-finite with recurrence 2000*n*(48911424697856946605*n -85862091501967897127)*(2*n+1) *(2*n-1)*(n+1)*a(n) +20*n*(2*n-1) *(9782284939571389321000*n^3 -124853950521493511435497*n^2 +291346534864358121613940*n -174094174192357320452243)*a(n-1) +6*(-1056620466555214160730036*n^5 +5240184994626612582867927*n^4 -10842595636486250859803566*n^3 +12555800263623324081669713*n^2 -8323849827256795107408998*n +2408908212964334471344960)*a(n-2) +(-11765946248792268093670721*n^5 +111908835475719217483707009*n^4 -409273054609037480568616913*n^3 +706828511197147489881004671*n^2 -556026097737885029117618846*n +145005575225258917734060720)*a(n-3) +12*(110108843793156901781209*n^5 -1706708924562157727758594*n^4 +10728825545391547292463142*n^3 -34121900584137543620498771*n^2+54762746448568812780284884*n -35381689886652975706836240)*a(n-4) -36*(3*n-11)*(n-4)*(3*n-13) *(2*n-7)*(36626509829570139536*n -97211536327074911575)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2024
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2/(1-x))^2)/x)
%o (PARI) a(n, s=2, t=2, u=-2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
%Y Cf. A211789, A369011.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jan 11 2024
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