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A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

%I A207569 #38 Feb 06 2021 18:10:02
%S A207569 1,1,3,18,151,1640,21825,343763,6253234,128993019,2975165831,
%T A207569 75866604098,2119310099700,64361149952242,2111222815441491,
%U A207569 74391641880144734,2802300974537717340,112379709083552152423,4780136025081921948194,214954914688567198802759
%N A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).
%C A207569 Compare g.f. to Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.
%C A207569 Compare g.f. to Sum_{n>=0} Product_{k=1..n} (1 - (1 - x)^(2*k-1)), which is the g.f. of A158691. - _Peter Bala_, Dec 04 2020
%H A207569 Michael De Vlieger, <a href="/A207569/b207569.txt">Table of n, a(n) for n = 0..200</a>
%H A207569 Hsien-Kuei Hwang and Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%F A207569 a(n) ~ sqrt(12) * 24^n * n^n / (exp(n+Pi^2/48) * Pi^(2*n+1)). - _Vaclav Kotesovec_, May 06 2014
%F A207569 G.f.: 1/2*( 1 + Sum_{n>=0} (1 + x)^(2*n+1) * Product_{k = 1..n} ((1 + x)^(2*k-1) - 1) ). Cf. A053250 and A215066. - _Peter Bala_, May 15 2017
%F A207569 Conjectural g.f.: Sum_{n>=0} (-1)^n*Product_{k = 1..n} 1 + ( -1/(1 + x) )^k. - _Peter Bala_, Dec 04 2020
%F A207569 From _Peter Bala_, Jan 29 2021: (Start)
%F A207569 Conjectural g.f.s: Sum_{n >= 0} (-1)^n*(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k*(1 + x)^k)^2. Also
%F A207569 (1/2)*( 1 + Sum_{n >= 0} 1/(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k/(1 + x)^k) ). (End)
%e A207569 G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 151*x^4 + 1640*x^5 + 21825*x^6 + ...
%e A207569 such that, by definition,
%e A207569 A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1) + ...
%t A207569 CoefficientList[Series[Sum[Product[(1+x)^(2*k-1)-1, {k, 1, n}], {n, 0, 20}], {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 06 2014 *)
%o A207569 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1) +x*O(x^n)),n)}
%o A207569 for(n=0,25,print1(a(n),", "))
%Y A207569 Cf. A179525, A207557, A207570, A207571, A215066, A053250, A158691.
%K A207569 nonn,easy
%O A207569 0,3
%A A207569 _Paul D. Hanna_, Feb 18 2012