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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 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; 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 July 14 03:42 EDT 2020. Contains 335716 sequences. (Running on oeis4.)