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.

1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, 5379, 19195, 71872, 281571, 1151338, 4902687, 21696505, 99598840, 473466698, 2327173489, 11810472444, 61808852380, 333170844940, 1847741027555, 10532499571707, 61649191750137, 370208647200165, 2278936037262610

Offset

1,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..300

Formula

G.f.: Sum_{k>=0} x^(2*k) / Product_{l=1..k} (1-l*x). - Ralf Stephan, Apr 18 2004

a(n) = Sum_{i=0..n} stirling2(n+1-i, i). - Zerinvary Lajos, Jan 31 2008

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

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

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

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

Example

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

Maple

with(combinat): seq(add(stirling2(n+1-i, i), i=0..n), n=1..26); # Zerinvary Lajos, Jan 31 2008

Mathematica

Table[Total[Table[StirlingS2[n - k + 1, k], {k, Floor[(n + 1)/2]}]], {n, 30}] (* T. D. Noe, Oct 29 2013 *)

Prog

(PARI) a(n) = sum(j=1, floor((n+1)/2), stirling(n+1-j, j, 2) ); /* Joerg Arndt, Apr 14 2013 */

Crossrefs

Row sums of A136011.

Sequence in context: A059019 A249560 A121953 * A171367 A092920 A177377

Adjacent sequences: A024424 A024425 A024426 * A024428 A024429 A024430

Keyword

nonn

Author

Clark Kimberling

Status

approved