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A202993 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^4)*x^n/n ), a power series in x with integer coefficients.

%I A202993 #16 Nov 01 2024 10:48:53
%S A202993 1,1,16,56,296,1052,4952,17292,70512,249712,931226,3212690,11399590,
%T A202993 38331770,130310820,428389292,1408697596,4524980036,14486512316,
%U A202993 45558807176,142488702483,439559056419,1347096766984,4082169772704,12286806024269,36629267989081
%N A202993 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^4)*x^n/n ), a power series in x with integer coefficients.
%C A202993 Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
%H A202993 Seiichi Manyama, <a href="/A202993/b202993.txt">Table of n, a(n) for n = 0..1000</a>
%F A202993 a(n) = (1/n)*Sum_{k=1..n} sigma(k^4) * a(n-k) for n>0, with a(0)=1.
%F A202993 log(a(n)) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (p*(1 + p + p^2 + p^4) / (p^5 - 1)) = 1.9202928959802946010362130828... - _Vaclav Kotesovec_, Nov 01 2024
%e A202993 G.f.: A(x) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...
%e A202993 log(A(x)) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 + 2801*x^7/7 + 8191*x^8/8 +...+ A202994(n)*x^n/n +...
%o A202993 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^4)*x^m/m)+x*O(x^n)), n)}
%o A202993 (PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^4)*a(n-k)))}
%Y A202993 Cf. A000203 (sigma), A000041 (partitions), A202994, A156304.
%K A202993 nonn
%O A202993 0,3
%A A202993 _Paul D. Hanna_, Dec 27 2011