A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

0, 1, 3, 9, 30, 112, 463, 2095, 10279, 54267, 306298, 1838320, 11677867, 78207601, 550277003, 4055549053, 31224520322, 250547144156, 2090779592827, 18110124715919, 162546260131455, 1509352980864191, 14478981877739094, 143299752100925452, 1461455003961745247

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..575

Formula

a(n) = Sum(k*binomial(n-1,k-1)*B(n-k), k=1..n) = Sum(k*A056857(n,k), k=1..n), where B(q) are the Bell numbers (A000110).

a(n) = (n-1)*B(n-1)+B(n). - Vladeta Jovovic, Nov 10 2006

a(n) ~ Bell(n) * (LambertW(n) + 1). - Vaclav Kotesovec, Jul 28 2021

Example

a(3)=9 because the 5 (=A000110(3)) set partitions of {1,2,3} are 123, 12|3, 13|2, 1|23 and 1|2|3 and 3+2+2+1+1=9.

Maple

with(combinat): seq(add(k*binomial(n-1, k-1)*bell(n-k), k=1..n), n=0..30);

Mathematica

Table[Sum[Binomial[n-1, k-1] * BellB[n-k] * k, {k, 1, n}], {n, 0, 22}] (* Geoffrey Critzer, Jun 14 2013 *)
Flatten[{0, Table[(n-1)*BellB[n-1] + BellB[n], {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 19 2016, after Vladeta Jovovic *)

Crossrefs

Cf. A000110, A056857.

Column p=1 of A270236 or of A270702.

Main diagonal of A270701.

Sequence in context: A117428 A339835 A134168 * A350589 A308554 A055730

Adjacent sequences: A124424 A124425 A124426 * A124428 A124429 A124430

Keyword

nonn

Author

Emeric Deutsch, Nov 10 2006

Extensions

a(0)=0 prepended by Alois P. Heinz, Mar 17 2016

Status

approved