A177257 -id:A177257 - cp's OEIS frontend

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

Emeric Deutsch, May 07 2010

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

			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;
    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;

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

Cf. A000110, A002378, A162148, A168444, A177255, A177256, A177257.

Cf. A302560.

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

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.
T(n, 0) = A168444(n).
Sum_{k=0..n} T(n, k) = A000110(n) (row sums).
Sum_{k=0..n} k*T(n, k) = A177255(n).
From G. C. Greubel, May 12 2024: (Start)
T(n, n) = 1.
T(n, n-1) = n-1, for n >= 1.
T(n, n-2) = A002378(n-2), for n >= 2.
T(n, n-3) = A162148(n-3), for n >= 3.
T(n, n-4) = A302560(n-3), for n >= 4. (End)

A177255 a(n) = Sum_{j=1..n} j*B(j-1), where B(k) = A000110(k) are the Bell numbers.

Original entry on oeis.org

0, 1, 3, 9, 29, 104, 416, 1837, 8853, 46113, 257583, 1533308, 9676148, 64452909, 451475027, 3314964857, 25442301577, 203604718076, 1695172374548, 14654631691569, 131309475792709, 1217516798735521, 11664652754184043, 115319114738472472, 1174967255260496776

Offset: 0

Emeric Deutsch, May 07 2010

nonn

Number of adjacent blocks in all partitions of the set {1,2,...,n}. An adjacent block is a block of the form (i, i+1, i+2, ...). Example: a(3)=9 because in 1-2-3, 1-23, 12-3, 13-2, and 123 we have 3, 2, 2, 1, and 1 adjacent blocks, respectively.

Harvey P. Dale, Table of n, a(n) for n = 0..575

Cf. A000110, A177254, A177256, A177257.

Partial sums of A052889.

[n eq 0 select 0 else (&+[j*Bell(j-1): j in [1..n]]): n in [0..30]]; // G. C. Greubel, May 11 2024

with(combinat): a := proc (n) options operator, arrow: sum(j*bell(j-1), j = 1 .. n) end proc; seq(a(n), n = 0 .. 23);

With[{nn=30},Join[{0},Accumulate[BellB[Range[0,nn-1]]Range[nn]]]] (* Harvey P. Dale, Nov 10 2014 *)

[sum(j*bell_number(j-1) for j in range(1,1+n)) for n in range(31)] # G. C. Greubel, May 11 2024

a(n) = Sum_{k=0..n} k * A177254(n,k).

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).

Original entry on oeis.org

1, 1, 2, 0, 4, 1, 8, 6, 1, 16, 25, 11, 32, 89, 77, 5, 64, 290, 433, 90, 128, 893, 2132, 951, 36, 256, 2645, 9602, 7710, 934, 512, 7618, 40589, 53137, 13790, 329, 1024, 21489, 163739, 328119, 152600, 11599, 2048, 59665, 637587, 1872748, 1409791, 228103

Offset: 0

Emeric Deutsch, May 07 2010

Row n contains 1+ floor(n/2) entries.
Sum of entries in row n = A000110(n) (the Bell numbers).
T(n,0) = 2^{n-1} (A000079).
Sum(k*a(n,k),k>=0) = A177257(n).

			T(4,1)=6 because we have 134-2, 124-3, 14-23, 1-24-3, 14-2-3, and 13-2-4.
Triangle starts:
1;
1;
2,0;
4,1;
8,6,1;
16,25,11;
32,89,77,5;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000079, A000110, A177254, A177255, A177257.

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

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.

cp's OEIS Frontend

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, ...).

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A177255 a(n) = Sum_{j=1..n} j*B(j-1), where B(k) = A000110(k) are the Bell numbers.

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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).

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