A386647 G.f. A(x) satisfies: A(x)^7 = A(x^7) / (1 - 7*x).

1, 1, 4, 20, 110, 638, 3828, 23515, 146968, 930797, 5957100, 38450370, 249927394, 1634140604, 10738638021, 70875009760, 469546933535, 3121106054760, 20807373517870, 139080864081230, 931841783576460, 6256651942091035, 42090203778813320, 283651372136401905, 1914646755015446620

Offset: 1

Author

Paul D. Hanna, Aug 11 2025

Keywords

• nonn

Comments

The EULER transform of A373277, where A373277 is the number of certain monic irreducible polynomials over GF(7).

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.

Compare g.f. to: H(x)^5 = H(x^5)/(1 - 5*x) where H(x) is the g.f. of A372535, the EULER transform of A054662.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146968*x^9 + 930797*x^10 + 5957100*x^11 + 38450370*x^12 +...
where A(x)^7 = A(x^7) / (1 - 7*x).
Also, when expressed as the EULER transform of A373277,
A(x) = x/( (1-x) * (1-x^2)^3 * (1-x^3)^16 * (1-x^4)^84 * (1-x^5)^480 * (1-x^6)^2792 * (1-x^7)^16807 * (1-x^8)^102900 * ... * (1-x^n)^A373277(n) * ... ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764808*x^15 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..700

Crossrefs

Cf. A123916, A271929, A372535, A373277.

Programs

PARI

{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^7)/(1 - 7*x +x*O(x^n)))^(1/7)); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))

PARI

/* EULER transform of A373277 */
{A373277(n) = 1/(7*n) * sumdiv(n, d, (gcd(d, 7)==1)*(moebius(d)*7^(n/d)))} \\ after Seiichi Manyama in A373277
{a(n) = my(A = x/prod(m=1, n, (1-x^m +x*O(x^n))^A373277(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)^7 = A(x^7) / (1 - 7*x).

(2) A(x) = x / Product_{n>=1} (1 - x^n)^A373277(n).

a(n) ~ c * 7^n / n^(6/7), where c = 0.02181670654997947129840613123487745678041711647162749305767393184541296... - Vaclav Kotesovec, Aug 12 2025