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%I #17 Mar 30 2012 18:37:32
%S 1,6,36,222,1446,10116,75924,602256,4958352,41783046,357442416,
%T 3091766904,26991194550,237605649780,2107693469880,18826297197444,
%U 169203629332230,1529098507275372,13885733651626548,126641707880226888,1159483975207373952,10652962589325269040,98187525261135608400
%N G.f. satisfies: A(x) = H(x*A(x)) where H(x) = A(x/H(x)) is the theta series of planar hexagonal lattice A_2 (A004016).
%F Let q = x*A(x), then g.f. A(x) satisfies:
%F (1) A(x) = 1 + 6*Sum_{n>=1} q^n/(1 + q^n + q^(2*n)).
%F (2) A(x) = 1 + 6*Sum_{n>=1} q^(3*n-2)/(1-q^(3*n-2)) - q^(3*n-1)/(1-q^(3*n-1)).
%F a(n) = [x^n] H(x)^(n+1)/(n+1) where H(x) is the g.f. of A004016.
%e G.f.: A(x) = 1 + 6*x + 36*x^2 + 222*x^3 + 1446*x^4 + 10116*x^5 +...
%e where the g.f. satisfies the series:
%e A(x) = 1 + 6*x*A(x)/(1 + x*A(x) + x^2*A(x)^2) + 6*x^2*A(x)^2/(1 + x^2*A(x)^2 + x^4*A(x)^4) + 6*x^3*A(x)^3/(1 + x^3*A(x)^3 + x^6*A(x)^6) +...
%e The g.f. satisfies: A(x) = H(x*A(x)) where H(x) = A(x/H(x)) begins:
%e H(x) = 1 + 6*x + 6*x^3 + 6*x^4 + 12*x^7 + 6*x^9 + 6*x^12 +...+ A004016(n)*x^n +...
%e so that A(x) = (1/x)*Series_Reversion(x/H(x)).
%e The coefficients in powers of H(x) begin:
%e H^1: [(1), 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0,...];
%e H^2: [1,(12), 36, 12, 84, 72, 36, 96, 180, 12, 216, 144, 84,...];
%e H^3: [1, 18,(108), 234, 234, 864, 756, 900, 1836, 2178, 1296,...];
%e H^4: [1, 24, 216, (888), 1752, 3024, 7992, 8256, 14040,...];
%e H^5: [1, 30, 360, 2190, (7230), 14976, 32760, 72060, 92520,...];
%e H^6: [1, 36, 540, 4356, 20556, (60696), 137916, 325152,...];
%e H^7: [1, 42, 756, 7602, 46914, 187488, (531468), 1302132,...];
%e H^8: [1, 48, 1008, 12144, 92784, 473760, 1706544, (4818048),...]; ...
%e in which the coefficients in parenthesis form initial terms of this sequence:
%e [1/1, 12/2, 108/3, 888/4, 7230/5, 60696/6, 531468/7, 4818048/8,...].
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+6*sum(k=1,n,x^k*A^k/(1+x^k*A^k+x^(2*k)*A^(2*k)+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+6*sum(k=1,n, (x*A)^(3*k-2)/(1-(x*A)^(3*k-2))-(x*A)^(3*k-1)/(1-(x*A)^(3*k-1)),x*O(x^n)));polcoeff(A,n)}
%Y Cf. A004016.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 19 2011
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