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%I #10 Mar 30 2012 18:37:33
%S 1,2,11,38,156,516,1991,6434,23065,75132,255335,816480,2724245,
%T 8570794,27763807,87057596,276339126,855374534,2681503010,8218321006,
%U 25421912010,77383062314,236519199902,714226056554,2165295121179,6490447624984,19503550719297,58127246438024
%N G.f.: exp( Sum_{n>=1} sigma(n)*A002203(n)*x^n/n ) where A002203 is the companion Pell numbers.
%C Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), and to the g.f. of Pell numbers: exp( Sum_{n>=1} A002203(n)*x^n/n ).
%F G.f.: Product_{n>=1} 1/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)).
%F a(n) = (1/n)*Sum_{k=1..n} sigma(k)*A002203(k)*a(n-k) for n>0, with a(0) = 1.
%e G.f.: A(x) = 1 + 2*x + 11*x^2 + 38*x^3 + 156*x^4 + 516*x^5 + 1991*x^6 +...
%e where
%e A(x) = 1/((1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...).
%e The companion Pell numbers (starting at offset 1) begin:
%e A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...].
%o (PARI) /* Subroutine used in PARI programs below: */
%o {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}
%o (PARI) {a(n)=polcoeff(prod(m=1, n, 1/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
%o (PARI) {a(n)=if(n<0,0,if(n==0,1,(1/n)*sum(k=1,n,sigma(k)*A002203(k)*a(n-k))))}
%Y Cf. A156234, A000129 (Pell), A002203 (companion Pell), A000203 (sigma).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 02 2012
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