A203801 G.f.: Product_{n>=1} (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

1, 1, 2, 7, 9, 27, 53, 109, 206, 463, 907, 1756, 3591, 6849, 13706, 27132, 51477, 99168, 195160, 366269, 707173, 1355524, 2558372, 4836092, 9186600, 17245564, 32428375, 61057276, 113946770, 212495896, 397836811, 737325660, 1368659832, 2544085015, 4694930535

Offset: 0

Author

Paul D. Hanna, Jan 11 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 into distinct parts.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 9*x^4 + 27*x^5 + 53*x^6 +...
where A(x) = (1+x-x^2) * (1+3*x^2+x^4) * (1+4*x^3-x^6) * (1+7*x^4+x^8) * (1+11*x^5-x^10) * (1+18*x^6+x^12) *...* (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) *...
and 1/A(x) = (1-x-x^2) * (1-4*x^3-x^6) * (1-11*x^5-x^10) * (1-29*x^7-x^14) * (1-76*x^9-x^18) * (1-199*x^11-x^22) *...* (1 - Lucas(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)) = x + 1*3*x^2/2 + 4*4*x^3/3 + 1*7*x^4/4 + 6*11*x^5/5 + 4*18*x^6/6 + 8*29*x^7/7 + 1*47*x^8/8 +...+ A000593(n)*Lucas(n)*x^n/n +...
		

Links

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

Crossrefs

Cf. A156234, A203860, A203850, A000204, A000593.

Programs

Mathematica

max = 40; s = Product[1 + LucasL[n]*x^n + (-1)^n*x^(2*n), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 14 2015 *)

PARI

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

PARI

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

PARI

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

Formula

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

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

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