OFFSET
0,3
COMMENTS
Partial sums of A000296.
a(n) is the total number of complete rhyme schemes for 0 to n lines; in other words, a(n) is the total number of rhyme schemes for 0 to n lines where each line rhymes with at least one other line.
FORMULA
G.f.: (G(0)-1)/(1-x) where G(k) = 1 + (1-x)/(1+x-x*k)/(1-x/(x+(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
G.f.: (1+x*sum{k>=0, x^k/prod[p=0..k, 1-p*x]})/(1-x^2). - Sergei N. Gladkovskii, Jan 25 2014
MATHEMATICA
m=30; CoefficientList[Series[(1+x*Sum[x^k/Product[1-p*x, {p, 0, k}], {k, 0, m}])/(1-x^2), {x, 0, m}], x] (* Georg Fischer, Aug 28 2020 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Anonymous, May 03 2009
EXTENSIONS
a(22)-a(23) corrected by Georg Fischer, Aug 28 2020
STATUS
approved