A372535 G.f. A(x) satisfies: A(x)^5 = A(x^5) / (1 - 5*x).

1, 1, 3, 11, 44, 185, 801, 3547, 15961, 72710, 334463, 1550679, 7236463, 33955573, 160075762, 757689991, 3599019810, 17148240314, 81930357294, 392402777679, 1883531191109, 9058879060004, 43647287768424, 210645440011836, 1018118905986455, 4927692357099550, 23880341433363005

Offset: 1

Author

Paul D. Hanna, May 30 2024

Keywords

• nonn

Comments

The EULER transform of A054662, where A054662 is the number of certain monic irreducible polynomials over GF(5).

Compare g.f. to: F(x)^2 = F(x^2)/(1 - 2*x) where F(x) is the g.f. of A123916, the EULER transform of A000048.

Compare g.f. to: G(x)^3 = G(x^3)/(1 - 3*x) where G(x) is the g.f. of A271929, the EULER transform of A046211.

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 801*x^7 + 3547*x^8 + 15961*x^9 + 72710*x^10 + 334463*x^11 + 1550679*x^12 +...
where A(x)^5 = A(x^5) / (1 - 5*x).
Also, when expressed as the EULER transform of A054662,
A(x) = x/( (1-x) * (1-x^2)^2 * (1-x^3)^8 * (1-x^4)^30 * (1-x^5)^125 * (1-x^6)^516 * (1-x^7)^2232 * (1-x^8)^9750 * ... * (1-x^n)^A054662(n) * ... ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 25*x^7 + 125*x^8 + 625*x^9 + 3126*x^10 + 15630*x^11 + 78150*x^12 + 390750*x^13 + 1953750*x^14 + 9768753*x^15 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..630

Crossrefs

Cf. A123916, A271929, A054662.

Programs

PARI

{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^5)/(1 - 5*x +x*O(x^n)))^(1/5)); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))

PARI

/* EULER transform of A054662 */
{A054662(n) = 1/(5*n) * sumdiv(n, d, if(gcd(d, 5)==1, moebius(d)*5^(n/d), 0 ) )} \\ after Joerg Arndt's program in A046211
{a(n) = my(A = x/prod(m=1, n, (1-x^m +x*O(x^n))^A054662(m))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^5 = A(x^5) / (1 - 5*x).

(2) A(x) = x / Product_{n>=1} (1 - x^n)^A054662(n).

a(n) ~ c * 5^n / n^(4/5), where c = 0.04356776732312620727955274802792860524970647403648680057626... - Vaclav Kotesovec, Jun 01 2024