%I #7 Mar 26 2022 02:43:43

%S 1,3,13,87,817,9819,143029,2442783,47817913,1054997475,25895101885,

%T 699790692519,20644163034049,660099532324971,22739373410768581,

%U 839552217608213295,33071685749731393225,1384473468760664408307

%N Partial sums of A004123.

%C 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.

%H G. C. Greubel, <a href="/A174278/b174278.txt">Table of n, a(n) for n = 1..390</a>

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

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

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

%t A004123[n_]:= A004123[n]= Sum[2^k*k!*StirlingS2[n-1,k], {k,0,n-1}];

%t A174278[n_]:= Sum[A004123[j], {j,0,n}];

%t Table[A174278[n], {n,30}] (* _G. C. Greubel_, Mar 25 2022 *)

%o (Sage)

%o def A004123(n): return sum(stirling_number2(n-1, k)*(2^k)*factorial(k) for k in (0..n-1))

%o def A174278(n): return sum(A004123(j) for j in (0..n))

%o [A174278(n) for n in (1..30)] # _G. C. Greubel_, Mar 25 2022

%Y Cf. A004123.

%K easy,nonn

%O 1,2

%A _Jonathan Vos Post_, Mar 15 2010