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A207081 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + (1+x)^k). 2

%I

%S 1,2,5,14,44,151,560,2221,9353,41575,194148,948716,4834965,25624951,

%T 140886544,801808675,4714489141,28590416466,178551890345,

%U 1146748103103,7564646759295,51195535619574,355096311786622,2521828180324820,18321335891780843,136055733744848751

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

%H Vaclav Kotesovec, <a href="/A207081/b207081.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: Sum_{n>=0, k=0..n*(n+1)/2} A053632(n,k)*x^n*(1+x)^k, where A053632(n,k) = number of partitions of k into distinct parts <= n.

%F G.f.: 1/(G(0) - 2*x) where G(k) = 1 + x + x*(1 + x)^k - x*(1 + (1 + x)^(k+1))/G(k+1); (recursively defined continued fraction; G(0)=2*x). - _Sergei N. Gladkovskii_, Dec 15 2012

%F G.f.: Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / ( Product_{k=0..n} 1 - x*(1+x)^k ). - _Paul D. Hanna_, Nov 09 2020

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 44*x^4 + 151*x^5 + 560*x^6 +...

%e such that, by definition,

%e A(x) = 1 + x*(1 + (1+x)) + x^2*(1 + (1+x))*(1 + (1+x)^2) + x^3*(1 + (1+x))*(1 + (1+x)^2)*(1 + (1+x)^3) +...

%o (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(k=1,m,(1+(1+x)^k)+x*O(x^n))),n)}

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

%Y Cf. A053632.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 19 2012

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Last modified August 7 21:08 EDT 2022. Contains 355994 sequences. (Running on oeis4.)