A177254 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks (0 <= k <= n). An adjacent block is a block of the form (i, i+1, i+2, ...).

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 1, 4, 6, 3, 1, 5, 13, 17, 12, 4, 1, 21, 51, 61, 44, 20, 5, 1, 91, 219, 255, 185, 90, 30, 6, 1, 422, 1019, 1182, 867, 440, 160, 42, 7, 1, 2103, 5108, 5964, 4430, 2322, 896, 259, 56, 8, 1, 11226, 27448, 32373, 24406, 13118, 5292, 1638, 392, 72, 9, 1

Offset

0,8

Comments

Row n contains n+1 entries.

Sum of entries in row n = A000110(n) (the Bell numbers).

T(n,0)=A168444(n).

Sum(k*a(n,k),k=0..n) = A177255(n).

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

The row generating polynomial P[n](t) is given by P[n](t)=Q[n](1,t,t), where Q[n](u,v,w) is obtained recursively from Q[n](u,v,w) =u(dQ[n-1]/du)_{w=v} + u(dQ[n-1]/dv)_{w=v} + w(dQ[n-1]/dw) + w(Q[n-1])_{w=v}, Q[0]=1. Here Q[n](u,v,w) is the trivariate generating polynomial of the partitions of {1,2,...,n}, where u marks blocks that are not adjacent, v marks adjacent blocks not ending with n, and w marks adjacent blocks ending with n.

Example

T(4,2)=6 because we have 1-234, 12-34, 123-4, 13-2-4, 14-2-3, and 1-24-3.
Triangle starts:
1;
0,1;
0,1,1;
0,2,2,1;
1,4,6,3,1;
5,13,17,12,4,1;
21,51,61,44,20,5,1;

Maple

Q[0] := 1: for n to 10 do Q[n] := expand(u*subs(w = v, diff(Q[n-1], u))+u*subs(w = v, diff(Q[n-1], v))+w*(diff(Q[n-1], w))+w*subs(w = v, Q[n-1])) end do: for n from 0 to 10 do P[n] := sort(expand(subs({v = t, w = t, u = 1}, Q[n]))) end do; for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A000110, A168444, A177255, A177256, A177257.

Sequence in context: A284992 A191687 A322190 * A340910 A132311 A254414

Adjacent sequences: A177251 A177252 A177253 * A177255 A177256 A177257

Keyword

nonn,tabl

Author

Emeric Deutsch, May 07 2010

Status

approved