A129355 - OEIS

%I #3 Mar 30 2012 18:37:03

%S 1,-2,-2,4,-1,12,-26,38,-51,6,98,-190,138,60,132,-1296,2990,-3738,

%T 3350,-3752,4077,1194,-12272,18528,-14848,9018,-2002,5644,-86729,

%U 290596,-514158,611070,-603150,657792,-952808,1406568,-1208636,-635286,3507362,-5062866,3791614

%N G.f.: A(x) = Product_{n>=1} [ (1-x)^2*(1 + 2x + 3x^2 +...+ n*x^(n-1)) ].

%C a(k) == 1 (mod 2) at k = 4*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,....

%F G.f.: A(x) = Product_{n>=1} ( 1 - (n+1)*x^n + n*x^(n+1) ) . G.f.: A(x) = Product_{n>=1} [ (1-x)*(1 + x + x^2 +...+ x^(n-1) - n*x^n) ] .

%e A(x) = (1 - 2x + x^2)(1 - 3x^2 + 2x^3)(1 - 4x^3 + 3x^4)(1 - 5x^4 + 4x^5)*...

%e Terms are even except at positions given by:

%e a(n) == 1 (mod 2) at n = [0, 4, 8, 20, 28, 48, 60, 88,...,4*A001318(n),...].

%o (PARI) a(n)=if(n==0,1,polcoeff(prod(k=1,n,1-(k+1)*x^k+k*x^(k+1)+x*O(x^n)),n))

%Y Cf. A129356, A129357, A129358.

%K sign

%O 0,2

%A _Paul D. Hanna_, Apr 10 2007