0,5

Half the number of (n-2)-element subsets of {1,...,n} with odd sum of the elements.

This is half the antepenultimate column of A159916, cf. formula.

The number of subsets of {1,...,n} with n-2 elements, adding up to an odd integer, is always even (cf. examples), so we divide it by 2.

We prefer to include a(0)=a(1)=a(2)=0, even if it might seem more natural to start only at n=2 or n=3.

From the rational g.f. it can be seen that the sequence is a linear recurrence with constant coefficients (3,-5,7,-7,5,-3,1) of order 7.

G.f.: x^3(1-x+x^2)/(1-3x+5x^2-7x^3+7x^4-5x^5+3x^6-x^7)

a(n)=3a(n-1)-5a(n-2)+7a(n-3)-7a(n-4)+5a(n-5)-3a(n-6)+a(n-7) for n>7

For n>2, a(n) = A159916(n(n-1)/2+n-2)/2 = T[n,n-2]/2 as defined there.

a(n)=[(n+1)/4][n/2] ; a(2n+1)=A093005(n) ; a(2n)=A093353(n-1)=[n/2]*n. - M. F. Hasler, May 03 2009

a(0)=a(1)=0 since there are no subsets with -2 or -1 elements.

a(2)=0 since the sum of the elements of a 0-element subset is zero.

a(3)=1 since for n=3 we have two singleton subsets of {1,2,3}, {1} and {3}, with odd sum of elements.

a(4)=2 since for n=4 we have four 2-element subsets of {1,2,3,4} with odd sum: {1,2}, {2,3}, {1,4}, {3,4}.

(PARI) A159915(n)=polcoeff((1-x+x^2)/(1-3*x+5*x^2-7*x^3+7*x^4-5*x^5+3*x^6-x^7)+O(x^(n-2\ )), n-3)

a(n, t=[0, 0, 0, 1, 2, 2, 3], c=[1, -3, 5, -7, 7, -5, 3]~)=while(n-->5, t=concat(vecextract(t, "^1"), t*c)); t[n+2] /* NB: a(n+1, [0, 0, 0, 0, 1, 2, 2]) gives the same result as a(n) */

(PARI) A159915(n)=(n+1)\4*(n\2) \\\\ M. F. Hasler, May 03 2009

Sequence in context: A106369 A032062 A011141 this_sequence A007801 A077871 A035587

Adjacent sequences: A159912 A159913 A159914 this_sequence A159916 A159917 A159918

easy,nonn

M. F. Hasler (MHasler(AT)univ-ag.fr), May 01 2009, May 03 2009