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

1, 1, 16, 56, 296, 1052, 4952, 17292, 70512, 249712, 931226, 3212690, 11399590, 38331770, 130310820, 428389292, 1408697596, 4524980036, 14486512316, 45558807176, 142488702483, 439559056419, 1347096766984, 4082169772704, 12286806024269, 36629267989081

Offset: 0

Author

Paul D. Hanna, Dec 27 2011

Keywords

• nonn

Comments

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.

Examples

			G.f.: A(x) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...
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 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A202994, A156304.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n, sigma(m^4)*x^m/m)+x*O(x^n)), n)}

PARI

{a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^4)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k^4) * a(n-k) for n>0, with a(0)=1.

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