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A369264 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^2)^3 ).

%I A369264 #20 Feb 15 2024 04:20:15
%S A369264 1,3,18,127,993,8268,71888,645087,5929527,55544315,528319662,
%T A369264 5088941628,49539243900,486606281496,4816930145376,48005470976271,
%U A369264 481262635723491,4850084768085465,49107197378659262,499298960719688343,5095861705240094097
%N A369264 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^2)^3 ).
%H A369264 P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H A369264 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369264 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k).
%F A369264 D-finite with recurrence +18*n*(3*n+2)*(2*n+3)*(3*n+1) *(2355222972296552964811*n -2353391098681877598217) *(n+1)*a(n) +3*n*(10232370941059360726949011*n^5 -6279411058144420889732231*n^4 +26515854213844281466097465*n^3 -21761373746876376187551525*n^2 -12108806260534489559295636*n +3394771165638813123794516)*a(n-1) +2*(-132629080888282243656059365*n^6 +156440924520330612537351287*n^5 -1546737637908414661531599805*n^4 +6858652031514251350543113065*n^3 -10688884261686986291502236950*n^2 +6884443241518652198616376568*n -1531720470240397832109679200)*a(n-2) +16*(-488032865226571716800174339*n^6 +5743512241166673419623793625*n^5 -28798925871340480498482300305*n^4 +76975939990931613139744649055*n^3 -114305622490237072905660442676*n^2 +89044784395613178550071941760*n -28430479725567026023998437760)*a(n-3) +384*(3*n-7) *(3*n-8)*(17416466042177225377415141*n^4 -183745766144088004186571330*n^3 +680994833213916542429809801*n^2 -1015881953145852406207817800*n +470197111913757817462248180)*a(n-4) +9216*(n-4)*(3*n-7)*(3*n-10) *(85246481204976073615097*n -71936955710157680798041)*(3*n-8) *(3*n-11)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2024
%F A369264 a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^2)^3 )^(n+1). - _Seiichi Manyama_, Feb 14 2024
%p A369264 A369264 := proc(n)
%p A369264     add(binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k),k=0..floor(n/2)) ;
%p A369264     %/(n+1) ;
%p A369264 end proc;
%p A369264 seq(A369264(n),n=0..70) ; # _R. J. Mathar_, Jan 25 2024
%o A369264 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^2)^3)/x)
%o A369264 (PARI) a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
%Y A369264 Cf. A369262, A369263.
%Y A369264 Cf. A365843, A369270.
%K A369264 nonn
%O A369264 0,2
%A A369264 _Seiichi Manyama_, Jan 18 2024