A227682 G.f.: exp( Sum_{n>=1} x^n / (n*(1-x)^n * (1-x^n)) ).
1, 1, 3, 7, 16, 35, 76, 162, 342, 715, 1484, 3060, 6278, 12824, 26102, 52969, 107224, 216601, 436798, 879584, 1769117, 3554726, 7136736, 14318524, 28711315, 57544864, 115290624, 230910993, 462362571, 925610398, 1852669016, 3707705019, 7419275371, 14844857959
Offset: 0
Keywords
Examples
From _Joerg Arndt_, May 01 2014: (Start) The a(5) = 35 compositions as described in the first comment are (here p:s stands for a part p of sort s) 01: [ 1:0 1:0 1:0 1:0 1:0 ] 02: [ 1:0 1:0 1:0 2:0 ] 03: [ 1:0 1:0 1:0 2:1 ] 04: [ 1:0 1:0 2:0 1:0 ] 05: [ 1:0 1:0 3:0 ] 06: [ 1:0 1:0 3:1 ] 07: [ 1:0 1:0 3:2 ] 08: [ 1:0 2:0 1:0 1:0 ] 09: [ 1:0 2:0 2:0 ] 10: [ 1:0 2:0 2:1 ] 11: [ 1:0 2:1 2:1 ] 12: [ 1:0 3:0 1:0 ] 13: [ 1:0 4:0 ] 14: [ 1:0 4:1 ] 15: [ 1:0 4:2 ] 16: [ 1:0 4:3 ] 17: [ 2:0 1:0 1:0 1:0 ] 18: [ 2:0 1:0 2:0 ] 19: [ 2:0 1:0 2:1 ] 20: [ 2:0 2:0 1:0 ] 21: [ 2:0 3:0 ] 22: [ 2:0 3:1 ] 23: [ 2:0 3:2 ] 24: [ 2:1 3:1 ] 25: [ 2:1 3:2 ] 26: [ 3:0 1:0 1:0 ] 27: [ 3:0 2:0 ] 28: [ 3:0 2:1 ] 29: [ 3:1 2:1 ] 30: [ 4:0 1:0 ] 31: [ 5:0 ] 32: [ 5:1 ] 33: [ 5:2 ] 34: [ 5:3 ] 35: [ 5:4 ] (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..500
Programs
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Mathematica
Flatten[{1,Table[SeriesCoefficient[Exp[Sum[x^k / (k*(1-x)^k * (1-x^k)),{k,1,n}]],{x,0,n}], {n,1,40}]}] (* Vaclav Kotesovec, May 01 2014 *) -
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/(1-x +x*O(x^n))^d/d) )), n)} for(n=0, 50, print1(a(n), ", "))
Formula
G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} 1/(d*(1-x)^d) ).
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 35*x^5 + 76*x^6 + 162*x^7 +...
where
log(A(x)) = x/((1-x)*(1-x)) + x^2/(2*(1-x)^2*(1-x^2)) + x^3/(3*(1-x)^3*(1-x^3)) + x^4/(4*(1-x)^4*(1-x^4)) + x^5/(5*(1-x)^5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 56*x^5/5 + 107*x^6/6 + 197*x^7/7 + 365*x^8/8 + 679*x^9/9 + 1280*x^10/10 +...
a(n) = A238350(n*(n+3)/2,n), a(n) is the number of compositions of n*(n+3)/2 with exactly n fixed points. - Alois P. Heinz, Apr 11 2014
a(n) ~ c * 2^n, where c = 1/(2*A048651) = 1.73137330972753180576... - Vaclav Kotesovec, May 01 2014
G.f.: Product {n >= 1} 1/(1 - x^n/(1 - x)). Row sums of A253829. - Peter Bala, Jan 20 2015
Comments