A177254 -id:A177254 - cp's OEIS frontend

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

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.

A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

Original entry on oeis.org

0, 0, 0, 1, 8, 47, 258, 1426, 8154, 48715, 305012, 2001719, 13754692, 98801976, 740584196, 5782218745, 46942426080, 395607218279, 3455493024350, 31236784338746, 291836182128670, 2814329123555051, 27980637362452980

Offset: 0

Emeric Deutsch, May 07 2010

nonn

Number of blocks not consisting of consecutive integers in all partitions of the set {1,2,...,n} (a singleton is considered a block of consecutive integers). Example: a(3)=1 because in 1-2-3, 1-23, 12-3, 13-2, and 123 only the block 13 does not consist of consecutive integers.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A177254, A177255, A177256.

A177257:= func< n | n eq 0 select 0 else (&+[(Binomial(n,j)-(j+1))*Bell(j): j in [0..n-1]]) >;
[A177257(n): n in [0..30]]; // G. C. Greubel, May 12 2024

with(combinat): a:= proc(n) add((binomial(n, j)-j-1)*bell(j), j = 0 .. n-1) end proc: seq(a(n), n = 0 .. 22);

Table[Sum[(Binomial[n,j]-j-1)BellB[j],{j,0,n-1}],{n,0,30}] (* Harvey P. Dale, Oct 15 2015 *)

def A177257(n): return sum((binomial(n,j) -(j+1))*bell_number(j) for j in range(n))
[A177257(n) for n in range(31)] # G. C. Greubel, May 12 2024

a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*Bell(j), where Bell(n) = A000110(n) are the Bell numbers.
a(n) = Sum_{k=0..floor(n/2)} k*A177256(n,k).
a(n) = A005493(n-1) - A177255(n).

A168444 Number of partitions of the set {1,2,...,n} such that no block is a sequence of consecutive integers (including 1-element blocks).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 21, 91, 422, 2103, 11226, 63879, 385691, 2461004, 16535820, 116628147, 861033654, 6637143698, 53297137552, 444940442553, 3854539901147, 34592812084693, 321125878230123, 3079144039478532, 30457076370822777, 310407099470429818, 3255972198123974137, 35114803641531204063

Offset: 0

Richard Stanley, Nov 25 2009

Some similar results appear in Klazar (see links).

			For n=5 the a(5) = 5 partitions are 13-245, 14-235, 24-135, 25-135, 35-124.

Richard Stanley, Enumerative Combinatorics, volume 1, second edition, Cambridge Univ Press, 2011, page 192, solution 111.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Martin Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.

Column k=0 of A177254.

b:= func< n | n eq 0 select 1 else (&+[(-1)^(n+j)*Binomial(j,n-j)*Bell(j): j in [Ceiling(n/2)..n]]) >;
A168444:= func< n | n eq 0 select 1 else b(n)-b(n-1) >;
[A168444(n): n in [0..30]]; // G. C. Greubel, May 12 2024

with(combinat): y:=sum(bell(n)*x^n,n=0..20): z:=(1-x)*subs(x=x*(1-x),y): taylor(z,x=0,21);

nn = 20; b := Sum[BellB[n] (x - x^2)^n, {n, 0, nn}]; CoefficientList[ Series[ (1-x) b, {x, 0, nn}], x] (* Geoffrey Critzer, Jun 01 2013 *)

b(n):=if n=0 then 1 else sum(binomial(k,n-k)*(-1)^(n-k)*belln(k),k,ceiling(n/2),n); a(n):=if n=0 then 1 else b(n)-b(n-1); /* Vladimir Kruchinin, Sep 09 2010 */

N=66;  x = 'x+O('x^N);
B = serlaplace(exp(exp(x)-1));
gf = (1-x)*subst(B,'x, x*(1-x));
Vec(gf) \\ Joerg Arndt, Jun 01 2013

@CachedFunction
def b(n): return 1 if (n==0) else sum((-1)^(n+j)*binomial(j,n-j)*bell_number(j) for j in range((n//2), n+1))
def A168444(n): return 1 if (n==0) else b(n) - b(n-1)
[A168444(n) for n in range(31)] # G. C. Greubel, May 12 2024

Ordinary g.f.: (1-x)F(x(1-x)), where F(x) = sum_{n>=0} B(n)x^n (the ordinary g.f. for the Bell numbers)

a(n) = b(n)-b(n-1), b(n) = if n=0 then 1 else sum(binomial(k,n-k)*(-1)^(n-k)*B(k),k=ceiling(n/2)..n). - Vladimir Kruchinin, Sep 09 2010

Added more terms, Joerg Arndt, Jun 01 2013

cp's OEIS Frontend

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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A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

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A168444 Number of partitions of the set {1,2,...,n} such that no block is a sequence of consecutive integers (including 1-element blocks).

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