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G.f.: A(x) = exp( Sum_{n>=1} sigma_n(2n)*x^n/n ), where sigma_n(2n) is the sum of the n-th powers of the divisors of 2*n.
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%I #6 Oct 31 2024 11:18:50

%S 1,3,15,120,1450,25383,591563,17156364,595635903,24023004840,

%T 1102221504614,56652798990909,3222918574782830,200989079661549750,

%U 13632214370613131094,998992560620311541814,78653794343072884416393

%N G.f.: A(x) = exp( Sum_{n>=1} sigma_n(2n)*x^n/n ), where sigma_n(2n) is the sum of the n-th powers of the divisors of 2*n.

%F a(n) = (1/n)*Sum_{k=1..n} sigma(2*k,k)*a(n-k) for n>0, with a(0) = 1.

%F a(n) ~ 2^n * n^(n-1). - _Vaclav Kotesovec_, Oct 31 2024

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 120*x^3 + 1450*x^4 + 25383*x^5 +...

%e log(A(x)) = 3*x + 21*x^2/2 + 252*x^3/3 + 4369*x^4/4 + 103158*x^5/5 +...

%e sigma(2n,n) = [3,21,252,4369,103158,3037530,106237176,4311810305,...].

%o (PARI) {a(n)=polcoeff(exp(sum(k=1,n,sigma(2*k,k)*x^k/k,x*O(x^n))),n)}

%o (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(2*k,k)*a(n-k)))}

%Y Cf. variant: A023881 (number of partitions in expanding space).

%Y Cf. A179504.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 08 2009