A174278 Partial sums of A004123.

1, 3, 13, 87, 817, 9819, 143029, 2442783, 47817913, 1054997475, 25895101885, 699790692519, 20644163034049, 660099532324971, 22739373410768581, 839552217608213295, 33071685749731393225, 1384473468760664408307

Offset: 1

Author

Jonathan Vos Post, Mar 15 2010

Keywords

• easy
• nonn

Comments

Partial sums of the number of generalized weak orders on n points. Equivalently, partial sums of the number of bipartitional relations on a set of cardinality n.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..390

Crossrefs

Cf. A004123.

Programs

Mathematica

A004123[n_]:= A004123[n]= Sum[2^k*k!*StirlingS2[n-1,k], {k,0,n-1}];
A174278[n_]:= Sum[A004123[j], {j,0,n}];
Table[A174278[n], {n,30}] (* G. C. Greubel, Mar 25 2022 *)

Sage

def A004123(n): return sum(stirling_number2(n-1, k)*(2^k)*factorial(k) for k in (0..n-1))
def A174278(n): return sum(A004123(j) for j in (0..n))
[A174278(n) for n in (1..30)] # G. C. Greubel, Mar 25 2022

Formula

a(n) = Sum_{i=1..n} A004123(i).

a(n) = Sum_{i=1..n} Sum_{k >= 0} (k^n*(2/3)^k)/3.

a(n) = Sum_{i=1..n} Sum_{k = 0..n} Stirling2(n,k)*(2^k)*k!.