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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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^(2k)/prod[l=1..k, 1-lx]}. - Ralf Stephan, Apr 18 2004

a(n) = sum(stirling2(n+1-i,i), i=0..n). - 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

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Last modified November 28 11:48 EST 2014. Contains 250342 sequences.