A105487 Number of partitions of {1...n} containing 5 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, 12, 56, 297, 1632, 9531, 58634, 379371, 2574254, 18276457, 135463074, 1046041114, 8399533370, 70013963418, 604840440328, 5407301690915, 49958478263502, 476403955991034, 4683463406478004, 47414166201239781, 493803423334040824, 5285548108715948453

Offset

7,2

Links

Table of n, a(n) for n=7..29.

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, 5), where c(n, k, 5) is the case r=5 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(8) = 2 because the partitions of {1,...,8} with 5 strings of 3 consecutive integers are 1234567/8, 1/2345678.

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: A105487 := proc(n) local k ; add(c(n, k, 5), k=1..n) ; end: for n from 7 to 30 do printf("%d, ", A105487(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]];
A105487[n_] := Sum[c[n, k, 5], {k, 1, n}];
Table[A105487[n], {n, 7, 30}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

Crossrefs

Cf. A105486, A105482.

Sequence in context: A084128 A044047 A296944 * A098453 A256150 A180073

Adjacent sequences: A105484 A105485 A105486 * A105488 A105489 A105490

Keyword

nonn

Author

Augustine O. Munagi, Apr 10 2005

Extensions

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

Status

approved