A204275 G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.

1, 2, 5, 26, 57, 222, 698, 2096, 6038, 19730, 58915, 169952, 516024, 1484958, 4397513, 13029558, 37094682, 106442928, 311875984, 879620854, 2522107990, 7229956352, 20398904648, 57543374566, 163053304047, 457604617760, 1283583473614, 3606627675050

Offset: 0

Author

Paul D. Hanna, Jan 13 2012

Keywords

• nonn

Comments

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.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 57*x^4 + 222*x^5 + 698*x^6 +...
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)) *...
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)) *...
Also, the logarithm of the g.f. equals the series:
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 +...
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...]
and form the logarithm of a g.f. for Pell numbers:
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 +...
		

Links

• Eric Weisstein's World of Mathematics, Euler Identity.

Crossrefs

Cf. A203801, A204270, A000129 (Pell), A002203 (companion Pell), A000593.

Programs

PARI

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI

{a(n)=polcoeff(prod(k=1,n,1+A002203(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}

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)}

PARI

/* Exponential form using sum of odd divisors of n: */
{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

Formula

G.f.: Product_{n>=1} 1/(1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).

G.f.: exp( Sum_{n>=1} A000593(n) * A002203(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.

a(n) = (1/n)*Sum_{k=1..n} A000593(k) * A002203(k)*a(n-k) for n>0, with a(0) = 1.