A105483 Number of partitions of {1...n} containing one string of 3 consecutive integers, counted within a block.

1, 2, 8, 32, 141, 672, 3451, 18962, 110882, 686866, 4489422, 30853656, 222276063, 1674067342, 13149209956, 107481488424, 912490408782, 8031867965568, 73181346933680, 689194657064660, 6699707386510583, 67143409071264516, 692926011957479445, 7356058078964945382

Offset: 3

Author

Augustine O. Munagi, Apr 10 2005

Keywords

• nonn

Examples

			a(5) = 8 because the partitions of {1,2,3,4,5} with one 3-string of consecutive integers are 1235/4, 1345/2, 15/234, 123/45, 12/345, 123/4/5, 1/234/5, 1/2/345.
		

Links

• Augustine O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005:3 (2005), 451-463.

Crossrefs

Cf. A105484, A105489, A105493.

Programs

Maple

c := proc(n,k,r) option remember ; local j ; if r =0 then add(binomial(n-j,j)*combinat[stirling2](n-j-1,k-1),j=0..floor(n/2)) ; else if r <0 or r > n-k-1 then RETURN(0) fi ; if n <1 then RETURN(0) fi ; if k <1 then RETURN(0) fi ; RETURN( c(n-1,k-1,r)+(k-1)*c(n-1,k,r)+c(n-2,k-1,r)+(k-1)*c(n-2,k,r) +c(n-1,k,r-1)-c(n-2,k-1,r-1)-(k-1)*c(n-2,k,r-1) ) ; fi ; end: A105483 := proc(n) local k ; add(c(n,k,1),k=1..n) ; end: for n from 3 to 26 do printf("%d, ",A105483(n)) ; od ; # R. J. Mathar, Feb 20 2007

Mathematica

S2[_, -1] = 0;
S2[n_, k_] = StirlingS2[n, k];
c [n_, k_, r_] := c[n, k, r] = Which[r == 0, Sum[Binomial[n - j, j]*S2[n - j - 1, k - 1], {j, 0, Floor[n/2]}], r < 0 || r > n - k - 1, 0, n < 1, 0, k < 1, 0, True, c[n - 1, k - 1, r] + (k - 1)*c[n - 1, k, r] + c[n - 2, k - 1, r] + (k - 1)*c[n - 2, k, r] + c[n - 1, k, r - 1] - c[n - 2, k - 1, r - 1] - (k - 1)*c[n - 2, k, r - 1]];
A105483[n_] := Sum[c[n, k, 1], {k, 1, n}];
Table[A105483[n], {n, 3, 26}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

Formula

a(n) = Sum_{k=1..n} c(n, k, 1), where c(n, k, 1) is the case r=1 of c(n, k, r) given by c(n, k, r)=c(n-1, k-1, r)+(k-1)c(n-1, k, r)+c(n-2, k-1, r)+(k-1)c(n-2, k, r)+c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)c(n-2, k, r-1), r=0, 1, .., n-k-1, k=1, 2, .., n-2r, c(n, k, 0) = Sum_{0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).

Extensions

More terms from R. J. Mathar, Feb 20 2007