A174278 Partial sums of A004123.

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

Offset

1,2

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

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!.

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 *)

Prog

(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

Crossrefs

Cf. A004123.

Sequence in context: A054420 A363656 A300701 * A352121 A001831 A196561

Adjacent sequences: A174275 A174276 A174277 * A174279 A174280 A174281

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 15 2010

Status

approved