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

Author

Emeric Deutsch, May 07 2010

Keywords

• nonn

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

Crossrefs

Cf. A000110, A177254, A177256, A177257.

Partial sums of A052889.

Programs

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

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 *)

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

Formula

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