A156234 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*A000204(n)*x^n/n ).

1, 1, 5, 10, 30, 63, 170, 355, 880, 1875, 4349, 9189, 20810, 43355, 95140, 198247, 424527, 875965, 1849535, 3781820, 7873167, 16005196, 32883560, 66390850, 135198990, 271051271, 546931398, 1090751095, 2183512495, 4329540830

Offset: 0

Author

Paul D. Hanna, Feb 06 2009

Keywords

• nonn

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),

and to the g.f. of Fibonacci numbers: exp( Sum_{n>=1} A000204(n)*x^n/n ) where A000204 is the Lucas numbers.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 + ...
log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + ...
Also, the g.f. equals the product:
A(x) = 1/((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)) * ...).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A225528, A000203 (sigma), A000204 (Lucas), A000041 (partitions), A000045.

Programs

Maple

N:= 100: # to get a(0) to a(N)
G:= exp(add(numtheory:-sigma(n)*lucas(n)*x^n/n,n=1..N)):
S:= series(G,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Dec 23 2015

PARI

{a(n)=polcoeff(exp(sum(k=1,n,sigma(k)*(fibonacci(k-1)+fibonacci(k+1))*x^k/k)+x*O(x^n)),n)}
for(n=0,40,print1(a(n),", "))

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(prod(m=1,n,1/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(n)*A000204(k)*a(n-k) for n>0, with a(0) = 1.

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

Logarithmic derivative yields A225528.