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A320670
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G.f.: 1 / [ Sum_{n>=0} (-1)^n * (2*n+1)*(9*n+1) * x^(n*(n+1)/2) ].
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2
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1, 30, 900, 26905, 804300, 24043500, 718749221, 21486074010, 642298264200, 19200672023385, 573979141313067, 17158360616809020, 512926895536596641, 15333283058934704460, 458368573399636228200, 13702332910236820263571, 409613437916178164869149, 12244861486043905536773460, 366044223488302308042741521, 10942416433364118043444939230
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OFFSET
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0,2
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COMMENTS
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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)].
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LINKS
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EXAMPLE
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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 + ...
such that
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) + ...
Note that the nonzero coefficients of 1/A(x) can be generated by
(1 - 27*x + 8*x^2)/(1 + x)^3 = 1 - 30*x + 95*x^2 - 196*x^3 + 333*x^4 + ...
RELATED SERIES.
The cube root of the g.f. is an integer series:
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 + ...
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PROG
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(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)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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