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G.f.: 1 / [ Sum_{n>=0} (-1)^n * (2*n+1)*(9*n+1) * x^(n*(n+1)/2) ]^(1/3).
2

%I #9 Feb 24 2019 02:03:20

%S 1,10,200,4635,115400,2989000,79413182,2147670780,58847999800,

%T 1628799414030,45443174184489,1276097550750560,36027161027614673,

%U 1021735788590470670,29088548540723602200,830905722688352466737,23803614419994729655422,683666066800944445647240,19680193160906254634961364,567666652900351449376374940

%N G.f.: 1 / [ Sum_{n>=0} (-1)^n * (2*n+1)*(9*n+1) * x^(n*(n+1)/2) ]^(1/3).

%C Limit a(n+1)/a(n) = 29.8937006274429170027521943552718162106155192278570861849...

%H Paul D. Hanna, <a href="/A320671/b320671.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + 10*x + 200*x^2 + 4635*x^3 + 115400*x^4 + 2989000*x^5 + 79413182*x^6 + 2147670780*x^7 + 58847999800*x^8 + 1628799414030*x^9 + ...

%e such that

%e 1/A(x)^3 = 1 - 30*x + 95*x^3 - 196*x^6 + 333*x^10 - 506*x^15 + 715*x^21 - 960*x^28 + 1241*x^36 - 1558*x^45 + 1911*x^55 - 2300*x^66 + ... + (-1)^n * (2*n+1)*(9*n+1) * x^(n*(n+1)/2) + ...

%e Note that the nonzero coefficients of 1/A(x)^3 can be generated by

%e (1 - 27*x + 8*x^2)/(1 + x)^3 = 1 - 30*x + 95*x^2 - 196*x^3 + 333*x^4 + ...

%e RELATED SERIES.

%e The cube of the g.f. A(x) equals the g.f. of A320670:

%e A(x)^3 = 1 + 30*x + 900*x^2 + 26905*x^3 + 804300*x^4 + 24043500*x^5 + 718749221*x^6 + 21486074010*x^7 + 642298264200*x^8 + 19200672023385*x^9 + ... + A320670(n)*x^n + ...

%o (PARI) {a(n) = my(A = 1/sum(m=0,sqrtint(2*n+1), (-1)^m * (2*m+1)*(9*m+1) * x^(m*(m+1)/2) +x*O(x^n))^(1/3)); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A320670.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 18 2018