A105486 Number of partitions of {1...n} containing 4 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

1, 2, 11, 50, 255, 1362, 7746, 46556, 294965, 1963865, 13703812, 99974851, 760824922, 6027441398, 49616033975, 423649629415, 3746306203604, 34259548971914, 323564415957687, 3152120868598090, 31638011553779137, 326825518800658174, 3471291152755614386

Offset

6,2

Links

Table of n, a(n) for n=6..28.

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

Formula

a(n) = Sum_{k=1..n} (c(n, k, 4), ), where c(n, k, 4) is the case r=4 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_{j= 0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).

Example

a(7) = 2 because the partitions of {1,...,7} with 4 strings of 3 consecutive integers are 123456/7, 1/234567.

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: A105486 := proc(n) local k ; add(c(n, k, 4), k=1..n) ; end: for n from 6 to 29 do printf("%d, ", A105486(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]];
A105486[n_] := Sum[c[n, k, 4], {k, 1, n}];
Table[A105486[n], {n, 6, 29}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

Crossrefs

Cf. A105485, A105487, A105490, A105494.

Sequence in context: A187000 A154415 A108851 * A357548 A137960 A018933

Adjacent sequences: A105483 A105484 A105485 * A105487 A105488 A105489

Keyword

nonn

Author

Augustine O. Munagi, Apr 10 2005

Extensions

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

Status

approved