cp's OEIS Frontend

A024427 S(n,1) + S(n-1,2) + S(n-2,3) + ... + S(n+1-k,k), where k = floor((n+1)/2) and S(i,j) are Stirling numbers of the second kind.

%I A024427 #43 Jul 11 2019 01:51:29
%S A024427 1,1,2,4,9,22,58,164,495,1587,5379,19195,71872,281571,1151338,4902687,
%T A024427 21696505,99598840,473466698,2327173489,11810472444,61808852380,
%U A024427 333170844940,1847741027555,10532499571707,61649191750137,370208647200165,2278936037262610
%N A024427 S(n,1) + S(n-1,2) + S(n-2,3) + ... + S(n+1-k,k), where k = floor((n+1)/2) and S(i,j) are Stirling numbers of the second kind.
%C A024427 a(n) is the number of ways to partition {1,2,...,n+1} into any number of blocks such that each block has at least 2 elements and the smallest 2 elements in each block are consecutive integers. - _Geoffrey Critzer_, Dec 02 2013
%H A024427 Alois P. Heinz, <a href="/A024427/b024427.txt">Table of n, a(n) for n = 1..300</a>
%F A024427 G.f.: Sum_{k>=0} x^(2*k) / Product_{l=1..k} (1-l*x). - _Ralf Stephan_, Apr 18 2004
%F A024427 a(n) = Sum_{i=0..n} stirling2(n+1-i, i). - _Zerinvary Lajos_, Jan 31 2008
%F A024427 G.f.: ((G(0) - 1)/(x-1)-x)/x^3 where G(k) =  1 - x/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F A024427 G.f.: 1/x^2/Q(0) - 1/x^2 where Q(k) = 1 - x^2/(1 - x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Apr 14 2013
%F A024427 G.f.: T(0)/(x^2*(1-x^2)) - 1/x^2, where T(k) = 1 - (k+1)*x^3/((k+1)*x^3 - (1 - x^2 - x*k)*(1 - x - x^2 - x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 29 2013
%F A024427 G.f.: 1/(Q(0)-x^2), where Q(k) = 1 - x*(k+1)/( 1 - x^2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 03 2013
%e A024427 a(5) = 9 because we have: {1,2,3,4,5,6}; {1,2,3,4},{5,6}; {1,2,3},{4,5,6}; {1,2},{3,4,5,6}; {1,2,5,6},{3,4}; {1,2,5},{3,4,6}; {1,2,6},{3,4,5}; {1,2,3,6},{4,5}; {1,2},{3,4},{5,6}. - _Geoffrey Critzer_, Dec 02 2013
%p A024427 with(combinat): seq(add(stirling2(n+1-i, i), i=0..n), n=1..26); # _Zerinvary Lajos_, Jan 31 2008
%t A024427 Table[Total[Table[StirlingS2[n - k + 1, k], {k, Floor[(n + 1)/2]}]], {n, 30}] (* _T. D. Noe_, Oct 29 2013 *)
%o A024427 (PARI) a(n) = sum(j=1,floor((n+1)/2), stirling(n+1-j,j,2) ); /* _Joerg Arndt_, Apr 14 2013 */
%Y A024427 Row sums of A136011.
%K A024427 nonn
%O A024427 1,3
%A A024427 _Clark Kimberling_