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A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

%I A322199 #13 Apr 11 2025 07:56:31
%S A322199 1,3,14,51,195,663,2345,7707,25744,82980,267812,846150,2676163,
%T A322199 8337189,25947281,80053128,246468551,754366239,2305139065,7014997404,
%U A322199 21317567297,64606020012,195557995054,590855420007,1783577678925,5377112705874,16199746640340,48763788775530,146712079122114,441146762285301,1326002750336702,3984148679940612,11967872331787643
%N A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).
%C A322199 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n + 1. - _Seiichi Manyama_, Apr 11 2025
%H A322199 Vaclav Kotesovec, <a href="/A322199/b322199.txt">Table of n, a(n) for n = 0..1000</a>
%H A322199 Seiichi Manyama, <a href="https://oeis.org/wiki/User:Seiichi_Manyama/Generalized_Euler_transform">Generalized Euler transform</a>.
%F A322199 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).
%F A322199 a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - _Vaclav Kotesovec_, Oct 04 2020
%F A322199 a(n) = Sum_{k=0..n} 2^k * A322210(k,n-k). - _Seiichi Manyama_, Apr 11 2025
%e A322199 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 A322199 such that
%e A322199 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 A322199 RELATED SERIES.
%e A322199 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 A322199 (PARI) {a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}
%o A322199 for(n=0,30, print1(a(n),", "))
%Y A322199 Cf. A322209, A322200, A322210.
%Y A322199 Cf. A000051, A266964.
%K A322199 nonn
%O A322199 0,2
%A A322199 _Paul D. Hanna_, Dec 01 2018