A206141 - OEIS

%I #8 Feb 17 2025 08:14:56

%S 1,1,3,8,26,67,216,555,1704,4538,13320,35376,103863,273792,783694,

%T 2101835,5905044,15745360,44132278,117267422,325136638,868034994,

%U 2379074541,6337238658,17347580484,46039358056,125056019725,332678989816,898361151760,2382959919616

%N G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - A002203(k)*x^k + (-1)^k*x^(2*k)), where A002203 is the companion Pell numbers.

%C Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).

%C As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - 2*u^k*x - (u^k*x)^2).

%e G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 67*x^5 + 216*x^6 + 555*x^7 +...

%e where

%e A(x) = 1 + x/(1-2*x-x^2) + x^2/((1-2*x-x^2)*(1-6*x^2+x^4)) + x^3/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)) + x^4/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)) + x^5/((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)) +...

%e The companion Pell numbers begin:

%e A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}

%o {A002203(n)=Pell(n-1)+Pell(n+1)}

%o {a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-A002203(k)*x^k+(-1)^k*x^(2*k)+x*O(x^n))),n)}

%o for(n=0,51,print1(a(n),", "))

%Y Cf. A002203 (Co.Pell), A206141.

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Feb 04 2012