A200378 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).

1, 6, 36, 222, 1446, 10116, 75924, 602256, 4958352, 41783046, 357442416, 3091766904, 26991194550, 237605649780, 2107693469880, 18826297197444, 169203629332230, 1529098507275372, 13885733651626548, 126641707880226888, 1159483975207373952, 10652962589325269040, 98187525261135608400

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

Let q = x*A(x), then g.f. A(x) satisfies:

(1) A(x) = 1 + 6*Sum_{n>=1} q^n/(1 + q^n + q^(2*n)).

(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)).

a(n) = [x^n] H(x)^(n+1)/(n+1) where H(x) is the g.f. of A004016.

Example

G.f.: A(x) = 1 + 6*x + 36*x^2 + 222*x^3 + 1446*x^4 + 10116*x^5 +...
where the g.f. satisfies the series:
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) +...
The g.f. satisfies: A(x) = H(x*A(x)) where H(x) = A(x/H(x)) begins:
H(x) = 1 + 6*x + 6*x^3 + 6*x^4 + 12*x^7 + 6*x^9 + 6*x^12 +...+ A004016(n)*x^n +...
so that A(x) = (1/x)*Series_Reversion(x/H(x)).
The coefficients in powers of H(x) begin:
H^1: [(1), 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0,...];
H^2: [1,(12), 36, 12, 84, 72, 36, 96, 180, 12, 216, 144, 84,...];
H^3: [1, 18,(108), 234, 234, 864, 756, 900, 1836, 2178, 1296,...];
H^4: [1, 24, 216, (888), 1752, 3024, 7992, 8256, 14040,...];
H^5: [1, 30, 360, 2190, (7230), 14976, 32760, 72060, 92520,...];
H^6: [1, 36, 540, 4356, 20556, (60696), 137916, 325152,...];
H^7: [1, 42, 756, 7602, 46914, 187488, (531468), 1302132,...];
H^8: [1, 48, 1008, 12144, 92784, 473760, 1706544, (4818048),...]; ...
in which the coefficients in parenthesis form initial terms of this sequence:
[1/1, 12/2, 108/3, 888/4, 7230/5, 60696/6, 531468/7, 4818048/8,...].

Prog

(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)}
(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)}

Crossrefs

Cf. A004016.

Sequence in context: A218991 A351056 A166748 * A085687 A242136 A129327

Adjacent sequences: A200375 A200376 A200377 * A200379 A200380 A200381

Keyword

nonn

Author

Paul D. Hanna, Nov 19 2011

Status

approved