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A322204 G.f.: exp( Sum_{n>=1} A322203(n)*x^n/n ), where A322203(n) is the coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

%I A322204 #36 Dec 12 2020 03:14:09
%S A322204 1,1,3,7,20,54,168,518,1702,5672,19413,67329,236994,842362,3022320,
%T A322204 10924142,39749219,145457241,534996370,1976582432,7332199623,
%U A322204 27298096431,101968071485,382033462335,1435270419582,5405847465772,20408264704999,77211968620103,292706146651697,1111698968597495,4229571286335997,16117966287887641,61515492682026560,235114188287816030,899821838980825557,3448133313264656915
%N A322204 G.f.: exp( Sum_{n>=1} A322203(n)*x^n/n ), where A322203(n) is the coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).
%C A322204 Conjecture: Euler transform of A003239. - _Georg Fischer_, Dec 10 2020
%H A322204 Alois P. Heinz, <a href="/A322204/b322204.txt">Table of n, a(n) for n = 0..1669</a> (first 401 terms from Paul D. Hanna)
%F A322204 a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1))) = 0.6176761088360252844346512553859... - _Vaclav Kotesovec_, Jun 18 2019, updated Aug 12 2019
%F A322204 G.f.: Product_{j>=1} c(x^j), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - _Alois P. Heinz_, Aug 24 2019
%e A322204 G.f.: 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 + ...
%e A322204 such that
%e A322204 log( A(x) ) = x + 5*x^2/2 + 13*x^3/3 + 45*x^4/4 + 131*x^5/5 + 497*x^6/6 + 1723*x^7/7 + 6525*x^8/8 + 24349*x^9/9 + 92655*x^10/10 + ... + A322203(n)*x^n/n + ...
%e A322204 Also,
%e A322204 A(x)^2  = 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 + ... + A322202(n)*x^n + ...
%p A322204 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A322204 b:= proc(n, i) option remember; `if`(n=0 or i=1, C(n),
%p A322204       add((t-> b(t, min(t, i-1)))(n-i*j)*C(j), j=0..n/i))
%p A322204     end:
%p A322204 a:= n-> b(n$2):
%p A322204 seq(a(n), n=0..35);  # _Alois P. Heinz_, Aug 24 2019
%t A322204 nmax = 25; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 12 2019 *)
%t A322204 nmax = 25; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 12 2019 *)
%o A322204 (PARI)
%o A322204 {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
%o A322204 {A322203(n) = polcoeff( n*polcoeff( L,n,x),n,y)}
%o A322204 {a(n) = polcoeff( exp( sum(m=1,n, A322203(m)*x^m/m ) +x*O(x^n) ),n) }
%o A322204 for(n=0,35, print1( a(n),", ") )
%Y A322204 Cf. A000108, A003239, A322200, A322203, A322202, A309682.
%K A322204 nonn
%O A322204 0,3
%A A322204 _Paul D. Hanna_, Nov 30 2018