A203861 - OEIS

%I #11 Mar 30 2012 18:37:34

%S 1,-3,-9,20,45,0,-151,-231,0,140,1107,2052,49,-1305,0,-15004,-28260,0,

%T 17710,0,81,324040,589953,0,-375570,-1089,0,-124124,-10659705,

%U -19764180,-121,12605358,0,0,4158315,0,567552368,1052295189,-780030,-669901660,0,0,-221399431,-85965,0

%N G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 where Lucas(n) = A000204(n).

%C a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

%H Paul D. Hanna, <a href="/A203861/b203861.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A000204(n) * x^n/n ).

%e G.f.: A(x) = 1 - 3*x - 9*x^2 + 20*x^3 + 45*x^4 - 151*x^6 - 231*x^7 +...

%e -log(A(x))/3 = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...

%e The g.f. equals the product:

%e A(x) = (1-x-x^2)^3 * (1-3*x^2+x^4)^3 * (1-4*x^3-x^6)^3 * (1-7*x^4+x^8)^3 * (1-11*x^5-x^10)^3 * (1-18*x^6+x^12)^3 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 *...

%e Positions of zeros form A020757:

%e [5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...]

%e which are numbers that are not the sum of two triangular numbers.

%o (PARI) /* Subroutine used in PARI programs below: */

%o {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n, -3*sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^3, n)}

%Y Cf. A203860, A203850, A156234, A020757.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 07 2012