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A204383 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 where A002203(n) is the companion Pell numbers. 3

%I

%S 1,-6,-9,70,90,0,-1411,-1722,0,490,60534,75222,49,-21510,0,-6067754,

%T -7542180,0,2156110,0,81,1420032740,1764323886,0,-504516870,-8118,0,

%U -50196874,-783087782910,-973096740630,-121,278263575996,0,0,27685627830,0,1024173639305948

%N G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 where A002203(n) is the companion Pell numbers.

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

%H Paul D. Hanna, <a href="/A204383/b204383.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A002203(n) * x^n/n ).

%e G.f.: A(x) = 1 - 6*x - 9*x^2 + 70*x^3 + 90*x^4 - 1411*x^6 - 1722*x^7 +...

%e -log(A(x))/3 = 1*2*x + 3*6*x^2/2 + 4*14*x^3/3 + 7*34*x^4/4 + 6*82*x^5/5 + 12*198*x^6/6 +...+ sigma(n)*A002203(n)*x^n/n +...

%e The g.f. equals the product:

%e A(x) = (1-2*x-x^2)^3 * (1-6*x^2+x^4)^3 * (1-14*x^3-x^6)^3 * (1-34*x^4+x^8)^3 * (1-82*x^5-x^10)^3 * (1-198*x^6+x^12)^3 *...* (1 - A002203(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,...].

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

%o {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n, -3*sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^3, n)}

%Y Cf. A203861, A204382, A204384, A020757.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 14 2012

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)