A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

1, 1, 3, 6, 12, 23, 43, 79, 142, 252, 442, 766, 1316, 2244, 3799, 6393, 10704, 17841, 29618, 49000, 80823, 132964, 218242, 357501, 584608, 954553, 1556575, 2535425, 4125805, 6708143, 10898897, 17696749, 28719276, 46586050, 75538702, 122444483, 198420445, 321461918

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d / d ).

G.f.: Product {n >= 1} 1/(1 - (1 + x)*x^n). - Peter Bala, Jan 20 2015

a(n) ~ phi^(n+1) / (sqrt(5)* A276987), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 16 2019

Example

G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 23*x^5 + 43*x^6 + 79*x^7 +...
where
log(A(x)) = x*(1+x)/(1-x) + x^2*(1+x)^2/(2*(1-x^2)) + x^3*(1+x)^3/(3*(1-x^3)) + x^4*(1+x)^4/(4*(1-x^4)) + x^5*(1+x)^5/(5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 17*x^4/4 + 26*x^5/5 + 38*x^6/6 + 57*x^7/7 + 81*x^8/8 + 118*x^9/9 + 180*x^10/10 +...

Mathematica

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*(1+x)^m/(1-x^m +x*O(x^n)) )), n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, (1+x +x*O(x^n))^d/d) )), n)}
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Cf. A160571, A227682, A306565, A306749.

Sequence in context: A181844 A162506 A328609 * A055244 A089068 A018180

Adjacent sequences: A227678 A227679 A227680 * A227682 A227683 A227684

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2013

Status

approved