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A363311 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^3 + A(x)^5).

%I A363311 #22 Dec 25 2024 05:34:03
%S A363311 1,2,16,180,2360,33760,510928,8043440,130371936,2161066432,
%T A363311 36465401344,624274702464,10816259970048,189305983870208,
%U A363311 3341924242051840,59437975940616960,1064030847809734144,19157066319365860352,346663014660754833408,6301645517153393121280
%N A363311 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^3 + A(x)^5).
%H A363311 Seiichi Manyama, <a href="/A363311/b363311.txt">Table of n, a(n) for n = 0..500</a>
%F A363311 G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A363311 (1) A(x) = 1 + x*(A(x)^3 + A(x)^5).
%F A363311 (2) A(x) = ((B(x) - 1)/x)^(1/5) where B(x) is the g.f. of A363310.
%F A363311 (3) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) for n >= 0.
%F A363311 D-finite with recurrence +8*n*(9639909229907389*n -4332180801077160)* (4*n+1) *(2*n-1) *(4*n-1) *a(n) +(-76286895522125418545*n^5 +381775644252842912682*n^4 -1033993649015194853931*n^3 +1551245138730960078498*n^2 -1139936487176542639744*n +315922393907140666080) *a(n-1) +2*(272671960126472445261*n^5 -3010900995907383509536*n^4 +12907236726784549786263*n^3 -27012522362058892089464*n^2 +27708850835094249342996*n -11174516509692301247280) *a(n-2) +4*(-627566489435411923*n^5 +144061968293307107646*n^4 -1706290600068411299693*n^3 +7720188970563268791354*n^2 -15561118085635458987024*n +11755034318370549299520) *a(n-3) -8*(n-4) *(696748847001815555*n^4 -19100265029551686306*n^3 +142472091583377235329*n^2 -415309555491080054458*n +422902881832258952040) *a(n-4) -96*(n-4) *(n-5)*(3*n-13) *(2465432947213573*n -7363340799047272) *(3*n-14) *a(n-5)=0. - _R. J. Mathar_, Jul 18 2023
%F A363311 a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - _Seiichi Manyama_, Apr 01 2024
%e A363311 G.f.: A(x) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + 130371936*x^8 + 2161066432*x^9 + 36465401344*x^10 + ...
%e A363311 where A(x) = 1 + x*(A(x)^3 + A(x)^5).
%e A363311 RELATED SERIES.
%e A363311 A(x)^3 = 1 + 6*x + 60*x^2 + 740*x^3 + 10200*x^4 + 150576*x^5 + 2328640*x^6 + 37242096*x^7 + ...
%e A363311 A(x)^5 = 1 + 10*x + 120*x^2 + 1620*x^3 + 23560*x^4 + 360352*x^5 + 5714800*x^6 + 93129840*x^7 + ... + A363310(n-1)*x^n + ...
%p A363311 A363311 := proc(n)
%p A363311     add(binomial(n,k)*binomial(3*n+2*k+1,n)/(3*n+2*k+1),k=0..n) ;
%p A363311 end proc:
%p A363311 seq(A363311(n),n=0..70) ; # _R. J. Mathar_, Jul 18 2023
%o A363311 (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) )}
%o A363311 for(n=0, 20, print1(a(n), ", "))
%Y A363311 Cf. A363309, A363310, A027307, A001764.
%K A363311 nonn
%O A363311 0,2
%A A363311 _Paul D. Hanna_, May 29 2023