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%I #16 Sep 23 2023 03:39:31
%S 0,2,22,140,680,2800,10304,34944,111360,337920,985600,2782208,7641088,
%T 20500480,53903360,139264000,354287616,889061376,2203975680,
%U 5404098560,13120307200,31569477632,75342282752,178467635200,419849830400
%N 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.
%C 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).
%F 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
%F Conjecture: G.f.:(-2*x*(x+1))/(2*x-1)^5 [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]
%e For n = 4, N = {1,2,3,4}, the 5 columns below give S sum(S) S' sum(S') prod(S):
%e { } 0 {1,2,3,4} 10 0
%e {1) 1 {2,3,4} 9 9
%e (2) 2 {1,3,4} 8 16
%e {3} 3 {1,2,4} 7 21
%e {4} 4 {1,2,3} 6 24
%e {1,2} 3 {3,4} 7 21
%e {1,3} 4 {2,4} 6 24
%e {1,4} 5 {2,3} 5 25
%e 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.
%o (PARI) print1(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); print1(s, ", ")); \\ _David Wasserman_, Dec 22 2004
%Y Cf. A000914.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Jan 02 2002, May 31 2003
%E More terms and formula from _John W. Layman_, Jan 05 2002
%E Further terms from _David Wasserman_, Dec 22 2004
%E Edited by _N. J. A. Sloane_, Nov 01 2008 at the suggestion of _R. J. Mathar_
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