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 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). 0
 1, 6, 36, 222, 1446, 10116, 75924, 602256, 4958352, 41783046, 357442416, 3091766904, 26991194550, 237605649780, 2107693469880, 18826297197444, 169203629332230, 1529098507275372, 13885733651626548, 126641707880226888, 1159483975207373952, 10652962589325269040, 98187525261135608400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS 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: A180218 A218991 A166748 * A085687 A242136 A129327 Adjacent sequences:  A200375 A200376 A200377 * A200379 A200380 A200381 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 19 2011 STATUS approved

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Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)