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

0,3

Comments

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.

Links

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

Formula

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

Maple

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

Mathematica

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

Prog

(Magma)
[n eq 0 select 0 else (&+[j*Bell(j-1): j in [1..n]]): n in [0..30]]; // G. C. Greubel, May 11 2024
(SageMath)
[sum(j*bell_number(j-1) for j in range(1, 1+n)) for n in range(31)] # G. C. Greubel, May 11 2024

Crossrefs

Cf. A000110, A177254, A177256, A177257.

Partial sums of A052889.

Sequence in context: A060719 A091152 A148945 * A136628 A151031 A151032

Adjacent sequences: A177252 A177253 A177254 * A177256 A177257 A177258

Keyword

nonn

Author

Emeric Deutsch, May 07 2010

Status

approved