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A177256 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that do not consist of consecutive integers (0<=k<=floor(n/2); a singleton is considered a block of consecutive integers).

%I A177256 #4 Aug 05 2015 14:30:43
%S A177256 1,1,2,0,4,1,8,6,1,16,25,11,32,89,77,5,64,290,433,90,128,893,2132,951,
%T A177256 36,256,2645,9602,7710,934,512,7618,40589,53137,13790,329,1024,21489,
%U A177256 163739,328119,152600,11599,2048,59665,637587,1872748,1409791,228103
%N A177256 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that do not consist of consecutive integers (0<=k<=floor(n/2); a singleton is considered a block of consecutive integers).
%C A177256 Row n contains 1+ floor(n/2) entries.
%C A177256 Sum of entries in row n = A000110(n) (the Bell numbers).
%C A177256 T(n,0) = 2^{n-1} (A000079).
%C A177256 Sum(k*a(n,k),k>=0) = A177257(n).
%H A177256 Alois P. Heinz, <a href="/A177256/b177256.txt">Rows n = 0..200, flattened</a>
%F A177256 The row generating polynomial P[n](u) is given by P[n](u)=Q[n](u,1,1), 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 do not consist of consecutive integers, v marks blocks consisting of consecutive integers and not ending with n, and w marks blocks consisting of consecutive integers and ending with n.
%e A177256 T(4,1)=6 because we have 134-2, 124-3, 14-23, 1-24-3, 14-2-3, and 13-2-4.
%e A177256 Triangle starts:
%e A177256 1;
%e A177256 1;
%e A177256 2,0;
%e A177256 4,1;
%e A177256 8,6,1;
%e A177256 16,25,11;
%e A177256 32,89,77,5;
%p A177256 Q[0] := 1: for n to 12 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 12 do P[n] := sort(expand(subs({v = 1, w = 1}, Q[n]))) end do: for n from 0 to 12 do seq(coeff(P[n], u, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A177256 Cf. A000079, A000110, A177254, A177255, A177257.
%K A177256 nonn,tabf
%O A177256 0,3
%A A177256 _Emeric Deutsch_, May 07 2010