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%I #17 Mar 06 2016 07:51:25
%S 1,1,9,251,16496,2083824,453803984,156304214576,80272385155584,
%T 58631012094472704,58713787327403063808,78225670182020153384448,
%U 135277046518915274471718912,297374407080303931562525442048,816367902369725640298981464096768
%N G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.
%H Vaclav Kotesovec, <a href="/A249593/b249593.txt">Table of n, a(n) for n = 0..180</a>
%F a(n) = Sum_{k=1..n} n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-3*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).
%F a(n) ~ c * n!^3, where c = Product_{k>=2} 1/(1-1/k^3) = 3*Pi/cosh(sqrt(3)*Pi/2) = 1.235488267746513477155075624616837... . - _Vaclav Kotesovec_, Mar 05 2016
%e G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 251*x^3/3!^3 + 16496*x^4/4!^3 +...
%e where
%e A(x) = 1/((1-x)*(1-x^2/2^3)*(1-x^3/3^3)*(1-x^4/4^3)*(1-x^5/5^3)*...).
%t Table[n!^3 * SeriesCoefficient[Product[1/(1 - x^m/m^3), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 05 2016 *)
%o (PARI) {a(n)=n!^3*polcoeff(prod(k=1, n, 1/(1-x^k/k^3 +x*O(x^n))),n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) /* Using logarithmic derivative: */
%o {b(k) = sumdiv(k, d, d^(1-3*k/d))}
%o {a(n) = if(n==0, 1, sum(k=1, n, n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k)))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A007841, A249588, A269791, A269793, A269794.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 02 2014
%E Name clarified by _Vaclav Kotesovec_, Mar 05 2016
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