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%I #14 Dec 12 2020 09:55:17
%S 1,2,7,20,63,190,613,1976,6604,22368,77270,270208,956780,3419212,
%T 12323226,44723840,163320766,599601984,2211844684,8193734760,
%U 30469278673,113692852342,425558528235,1597428832560,6011972255226,22680620270712,85754229105470,324898592591960,1233299357981416,4689870496585016,17863799895741982,68149300647823612,260364494604701847,996086232267182566,3815683108118138847,14634441964549504036
%N G.f.: exp( Sum_{n>=1} A322201(n)*x^n/n ), where A322201(n) is the coefficient of x^n*y^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).
%C Conjecture: Euler transform of A123611. - _Vaclav Kotesovec_, Dec 12 2020
%H Paul D. Hanna, <a href="/A322202/b322202.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) ~ c * 4^n / n^(3/2), where c = 4/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1)))^2 = 2.704933139869066452954644773467... - _Vaclav Kotesovec_, Jun 18 2019, updated Dec 12 2020
%F G.f.: Product_{j>=1} c(x^j)^2, where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - _Vaclav Kotesovec_, Dec 12 2020
%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 20*x^3 + 63*x^4 + 190*x^5 + 613*x^6 + 1976*x^7 + 6604*x^8 + 22368*x^9 + 77270*x^10 + 270208*x^11 + 956780*x^12 + ...
%e such that
%e log( A(x) ) = 2*x + 10*x^2/2 + 26*x^3/3 + 90*x^4/4 + 262*x^5/5 + 994*x^6/6 + 3446*x^7/7 + 13050*x^8/8 + 48698*x^9/9 + 185310*x^10/10 + ... + A322201(n)*x^n/n + ...
%e sqrt(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 54*x^5 + 168*x^6 + 518*x^7 + 1702*x^8 + 5672*x^9 + 19413*x^10 + 67329*x^11 + 236994*x^12 + ... + A322204(n)*x^n + ...
%t nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 12 2020 *)
%t nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 12 2020 *)
%o (PARI)
%o {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
%o {A322201(n) = polcoeff( 2*n*polcoeff( L,n,x),n,y)}
%o {a(n) = polcoeff( exp( sum(m=1,n, A322201(m)*x^m/m ) +x*O(x^n) ),n) }
%o for(n=0,35, print1( a(n),", ") )
%Y Cf. A322200, A322201, A322204.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 30 2018
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