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

0,3

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.

Links

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

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

Example

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 +...

Prog

(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)))}

Crossrefs

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

Sequence in context: A304692 A295986 A169882 * A221068 A187277 A316443

Adjacent sequences: A202990 A202991 A202992 * A202994 A202995 A202996

Keyword

nonn

Author

Paul D. Hanna, Dec 27 2011

Status

approved