This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A204275 #7 Feb 16 2025 08:33:16
%S A204275 1,2,5,26,57,222,698,2096,6038,19730,58915,169952,516024,1484958,
%T A204275 4397513,13029558,37094682,106442928,311875984,879620854,2522107990,
%U A204275 7229956352,20398904648,57543374566,163053304047,457604617760,1283583473614,3606627675050
%N A204275 G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.
%C A204275 Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions of distinct parts.
%H A204275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerIdentity.html">Euler Identity.</a>
%F A204275 G.f.: Product_{n>=1} 1/(1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).
%F A204275 G.f.: exp( Sum_{n>=1} A000593(n) * A002203(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.
%F A204275 a(n) = (1/n)*Sum_{k=1..n} A000593(k) * A002203(k)*a(n-k) for n>0, with a(0) = 1.
%e A204275 G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 57*x^4 + 222*x^5 + 698*x^6 +...
%e A204275 where A(x) = (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 A204275 and 1/A(x) = (1-2*x-x^2) * (1-14*x^3-x^6) * (1-82*x^5-x^10) * (1-478*x^7-x^14) * (1-2786*x^9-x^18) * (1-16238*x^11-x^22) *...* (1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...
%e A204275 Also, the logarithm of the g.f. equals the series:
%e A204275 log(A(x)) = 1*2*x + 1*6*x^2/2 + 4*14*x^3/3 + 1*34*x^4/4 + 6*82*x^5/5 + 4*198*x^6/6 + 8*478*x^7/7 + 1*1154*x^8/8 +...+ A000593(n)*A002203(n)*x^n/n +...
%e A204275 The companion Pell numbers (starting at offset 1) begin:
%e A204275 A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...]
%e A204275 and form the logarithm of a g.f. for Pell numbers:
%e A204275 log(1/(1-2*x-x^2)) = 2*x + 6*x^2/2 + 14*x^3/3 + 34*x^4/4 + 82*x^5/5 +...
%o A204275 (PARI) /* Subroutine used in PARI programs below: */
%o A204275 {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
%o A204275 (PARI) {a(n)=polcoeff(prod(k=1,n,1+A002203(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}
%o A204275 (PARI) {a(n)=polcoeff(1/prod(k=1,n,1-A002203(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)),n)}
%o A204275 (PARI) /* Exponential form using sum of odd divisors of n: */
%o A204275 {A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
%o A204275 {a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}
%Y A204275 Cf. A203801, A204270, A000129 (Pell), A002203 (companion Pell), A000593.
%K A204275 nonn
%O A204275 0,2
%A A204275 _Paul D. Hanna_, Jan 13 2012