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A067057 Let A(n) = {1,2,3,...n}. Let B(r) and C(n-r) be two subsets of A(n) having r and n-r elements respectively, such that B(r) U C(n-r) = A(n) and B and C are disjoint; then a(n) = sum of the products of all combination sums of elements of B and C for r =1 to n-1. 1
0, 2, 22, 140, 680, 2800, 10304, 34944, 111360, 337920, 985600, 2782208, 7641088, 20500480, 53903360, 139264000, 354287616, 889061376, 2203975680, 5404098560, 13120307200, 31569477632, 75342282752, 178467635200, 419849830400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In other words, consider 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).

LINKS

Table of n, a(n) for n=1..25.

FORMULA

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, Jan 05 2002

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

EXAMPLE

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.

PROG

(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)

CROSSREFS

Cf. A000914.

Sequence in context: A062180 A244719 A084399 * A202738 A286778 A232977

Adjacent sequences:  A067054 A067055 A067056 * A067058 A067059 A067060

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Jan 02 2002, May 31 2003

EXTENSIONS

More terms and formula from John W. Layman, Jan 05 2002

Further terms from David Wasserman, Dec 22 2004

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

STATUS

approved

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Last modified April 22 22:37 EDT 2019. Contains 322379 sequences. (Running on oeis4.)