%I #6 Oct 04 2020 10:41:31

%S 1,3,14,51,195,663,2345,7707,25744,82980,267812,846150,2676163,

%T 8337189,25947281,80053128,246468551,754366239,2305139065,7014997404,

%U 21317567297,64606020012,195557995054,590855420007,1783577678925,5377112705874,16199746640340,48763788775530,146712079122114,441146762285301,1326002750336702,3984148679940612,11967872331787643

%N G.f.: Product_{n>=1} 1/(1 - (2^n + 1)*x^n).

%H Vaclav Kotesovec, <a href="/A322199/b322199.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).

%F a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - _Vaclav Kotesovec_, Oct 04 2020

%e G.f.: A(x) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ...

%e such that

%e A(x) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).

%e RELATED SERIES.

%e log( A(x) ) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ... + A322209(n)*x^n/n + ...

%o (PARI) {a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A322209, A322200.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 01 2018