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A067057
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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.
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1
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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).
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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 (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]
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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.
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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)
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CROSSREFS
| Cf. A000914.
Sequence in context: A083833 A062180 A084399 * A202738 A123960 A091169
Adjacent sequences: A067054 A067055 A067056 * A067058 A067059 A067060
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 2002, May 31 2003
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EXTENSIONS
| 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
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