1,2

In other words, lonsider the set N = {1,2,3,...,n}; let S and S' be subsets of N such that S union S' is N. Define prod(S) = ( sum of members of S)*( sum of members of S'); then a(n) = sum of all possible prod(S).

For n>1, all listed values are given by a(n)=(2^(n-2))*s(n+1, n-1), where the s(n+1, n-1) are Stirling numbers of the first kind (A000914). - John W. Layman (layman(AT)math.vt.edu), Jan 05 2002

Conjecture: G.f.:(-2*x*(x+1))/(2*x-1)^5 [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]

For n = 4, N = {1,2,3,4}, the 5 columns below give S sum(S) S' sum(S') prod(S):

{ } 0 {1,2,3,4} 10 0

{1) 1 {2,3,4} 9 9

(2) 2 {1,3,4} 8 16

{3} 3 {1,2,4} 7 21

{4} 4 {1,2,3} 6 24

{1,2} 3 {3,4} 7 21

{1,3} 4 {2,4} 6 24

{1,4} 5 {2,3} 5 25

Hence a(4) = 1*(2 + 3 + 4) + 2*(1 + 3 + 4) + 3*(1 + 2 + 4) + 4*(1 + 2 + 3) + (1 + 2)*(3 + 4) + (1 + 3)*(2 + 4) + (1 + 4)*(2 + 3) = 140.

(PARI) print(0); LIMIT = 40; V = vector(LIMIT*(LIMIT + 1)/2); V[1] = 1; for (i = 2, LIMIT, forstep (j = i*(i - 1)/2, 1, -1, V[i + j] += V[j]); V[i]++; k = i*(i + 1)/2; s = sum(j = 1, (k - 1)\2, j*(k - j)*V[j]); if (!(k%2), s += k*k*V[k\2]/8); print(s)); (Wasserman)

Cf. A000914.

Sequence in context: A083833 A062180 A084399 this_sequence A123960 A091169 A000184

Adjacent sequences: A067054 A067055 A067056 this_sequence A067058 A067059 A067060

nonn

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 2002, May 31 2003

More terms and formula from John W. Layman (layman(AT)math.vt.edu), Jan 05 2002

Further terms from David Wasserman (wasserma(AT)spawar.navy.mil), Dec 22 2004

Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 01 2008 at the suggestion of R. J. Mathar