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Number of partitions of sets containing from 0 to n elements into blocks of at least 2 elements.
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%I #21 Aug 28 2020 06:16:31

%S 1,1,2,3,7,18,59,221,936,4361,22083,120336,700653,4333933,28345090,

%T 195233255,1411303635,10675375402,84276173439,692752181561,

%U 5917018378496,52416910416933,480786834535247,4559132648864256

%N Number of partitions of sets containing from 0 to n elements into blocks of at least 2 elements.

%C Partial sums of A000296.

%C 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.

%C If the restriction that the blocks of the partitions must have at least 2 elements is removed, then A005001 is obtained except for the first term of A005001.

%F 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

%F 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

%F 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

%t 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 *)

%Y Cf. A000296, A005001, A000110.

%K easy,nonn

%O 0,3

%A Anonymous, May 03 2009

%E a(22)-a(23) corrected by _Georg Fischer_, Aug 28 2020