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%I #13 Feb 24 2019 02:03:41
%S 1,30,900,26905,804300,24043500,718749221,21486074010,642298264200,
%T 19200672023385,573979141313067,17158360616809020,512926895536596641,
%U 15333283058934704460,458368573399636228200,13702332910236820263571,409613437916178164869149,12244861486043905536773460,366044223488302308042741521,10942416433364118043444939230
%N G.f.: 1 / [ Sum_{n>=0} (-1)^n * (2*n+1)*(9*n+1) * x^(n*(n+1)/2) ].
%C a(n) ~ c*d^n, where d = 29.893700627442917002752194355271816210615519227857086... and c = 1.0071619287873131103030753829058024570785462927254481177... such that Sum_{n>=0} (-1)^n * (2*n+1)*(9*n+1) / d^(n*(n+1)/2) = 0 and c = 2/[Sum_{n>=1} (-1)^(n-1) * n*(n+1)*(2*n+1)*(9*n+1) / d^(n*(n+1)/2)].
%H Paul D. Hanna, <a href="/A320670/b320670.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 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 + ...
%e such that
%e 1/A(x) = 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) 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 root of the g.f. is an integer series:
%e A(x)^(1/3) = 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 + ... + A320671(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))); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A320671.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 18 2018
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