login
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)).
11

%I #36 Dec 12 2020 03:14:09

%S 1,1,3,7,20,54,168,518,1702,5672,19413,67329,236994,842362,3022320,

%T 10924142,39749219,145457241,534996370,1976582432,7332199623,

%U 27298096431,101968071485,382033462335,1435270419582,5405847465772,20408264704999,77211968620103,292706146651697,1111698968597495,4229571286335997,16117966287887641,61515492682026560,235114188287816030,899821838980825557,3448133313264656915

%N 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 Conjecture: Euler transform of A003239. - _Georg Fischer_, Dec 10 2020

%H 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 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 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 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 such that

%e 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 Also,

%e 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 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, C(n),

%p add((t-> b(t, min(t, i-1)))(n-i*j)*C(j), j=0..n/i))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..35); # _Alois P. Heinz_, Aug 24 2019

%t 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 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 (PARI)

%o {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}

%o {A322203(n) = polcoeff( n*polcoeff( L,n,x),n,y)}

%o {a(n) = polcoeff( exp( sum(m=1,n, A322203(m)*x^m/m ) +x*O(x^n) ),n) }

%o for(n=0,35, print1( a(n),", ") )

%Y Cf. A000108, A003239, A322200, A322203, A322202, A309682.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 30 2018