This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177254 #8 May 12 2024 02:00:06
%S A177254 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,
%T A177254 219,255,185,90,30,6,1,422,1019,1182,867,440,160,42,7,1,2103,5108,
%U A177254 5964,4430,2322,896,259,56,8,1,11226,27448,32373,24406,13118,5292,1638,392,72,9,1
%N 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, ...).
%C A177254 Sum of entries in row n = A000110(n) (the Bell numbers).
%H A177254 Alois P. Heinz, <a href="/A177254/b177254.txt">Rows n = 0..140, flattened</a>
%F A177254 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.
%F A177254 T(n, 0) = A168444(n).
%F A177254 Sum_{k=0..n} T(n, k) = A000110(n) (row sums).
%F A177254 Sum_{k=0..n} k*T(n, k) = A177255(n).
%F A177254 From _G. C. Greubel_, May 12 2024: (Start)
%F A177254 T(n, n) = 1.
%F A177254 T(n, n-1) = n-1, for n >= 1.
%F A177254 T(n, n-2) = A002378(n-2), for n >= 2.
%F A177254 T(n, n-3) = A162148(n-3), for n >= 3.
%F A177254 T(n, n-4) = A302560(n-3), for n >= 4. (End)
%e A177254 T(4,2)=6 because we have 1-234, 12-34, 123-4, 13-2-4, 14-2-3, and 1-24-3.
%e A177254 Triangle starts:
%e A177254 1;
%e A177254 0, 1;
%e A177254 0, 1, 1;
%e A177254 0, 2, 2, 1;
%e A177254 1, 4, 6, 3, 1;
%e A177254 5, 13, 17, 12, 4, 1;
%e A177254 21, 51, 61, 44, 20, 5, 1;
%e A177254 91, 219, 255, 185, 90, 30, 6, 1;
%e A177254 422, 1019, 1182, 867, 440, 160, 42, 7, 1;
%e A177254 2103, 5108, 5964, 4430, 2322, 896, 259, 56, 8, 1;
%p A177254 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
%Y A177254 Cf. A000110, A002378, A162148, A168444, A177255, A177256, A177257.
%Y A177254 Cf. A302560.
%K A177254 nonn,tabl
%O A177254 0,8
%A A177254 _Emeric Deutsch_, May 07 2010