cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, May 30 2024

Keywords

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

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

A372870 G.f. A(x) satisfies A(x)^3 = A(x^3) / (1 - 3*x)^3 with A(0)=1.

Original entry on oeis.org

1, 3, 9, 28, 84, 252, 758, 2274, 6822, 20471, 61413, 184239, 552729, 1658187, 4974561, 14923714, 44771142, 134313426, 402940361, 1208821083, 3626463249, 10879389971, 32638169913, 97914509739, 293743529832, 881230589496, 2643691768488, 7931075307172
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2024

Keywords

Comments

Euler transform of 3 * A046211(n).

Examples

			A(x)^3 = 1 + 9*x + 54*x^2 + 273*x^3 + 1242*x^4 + 5265*x^5 + 21231*x^6 + ... .

Links

Crossrefs

Cf. A372956, A372957.
Cf. A046211, A271929.

Programs

  • PARI
    b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
my(N=30, x='x+O('x^N)); Vec(1/prod(k=1, N, (1 - x^k)^b(k, 3))^3)

Formula

G.f.: A(x) = 1 / ( Product_{k>=1} (1 - x^k)^A046211(k) )^3.

A054666 Number of 6-ary Lyndon words with trace 1 mod 6.

Original entry on oeis.org

1, 3, 12, 54, 259, 1296, 6665, 34992, 186624, 1007769, 5496925, 30233088, 167444795, 932906715, 5224277604, 29386561536, 165947641615, 940369969152, 5345260877285, 30467987000514, 174102782860140, 997134120017175
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 6-ary Lyndon words with trace 5 mod 6.

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054660, etc.

Extensions

More terms from James Sellers, Apr 19 2000

A054663 Number of monic irreducible polynomials over GF(5) with zero trace.

Original entry on oeis.org

1, 2, 8, 30, 124, 516, 2232, 9750, 43400, 195248, 887784, 4068740, 18780048, 87191964, 406900992, 1907343750, 8975758272, 42385503300, 200773540296, 953674218720, 4541306267856, 21674415838068, 103660251783288
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 5-ary Lyndon words with trace 0 mod 5; also number of Lyndon words of trace 0 over GF(5). - Frank Ruskey and Nate Kube, Sep 11 2002

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054661, etc.

Extensions

More terms from James Sellers, Apr 19 2000

A054664 Number of 4-ary Lyndon words of length n with trace 0 mod 4.

Original entry on oeis.org

1, 1, 5, 14, 51, 165, 585, 2032, 7280, 26163, 95325, 349350, 1290555, 4792905, 17895679, 67106816, 252645135, 954429840, 3616814565, 13743869130, 52357696365, 199911109725, 764877654105, 2932030657200, 11258999068416
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 4-ary Lyndon words of length n with trace 2 mod 4.

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054660, etc.

Programs

  • Mathematica
    a[n_] := 1/(4 n) Sum[GCD[d, 4] MoebiusMu[d] 4^(n/d), {d, Divisors[n]}];
Array[a, 30] (* Andrey Zabolotskiy, Dec 19 2020 *)

Formula

From Andrey Zabolotskiy, Dec 19 2020: (Start)
a(n) = A068596(n) + A074403(n) + A074404(n) + A074405(n).
a(n) = A074410(n) + A074411(n) + A074412(n) + A074413(n). (End)

Extensions

More terms from James Sellers, Apr 19 2000

A054665 Number of 6-ary Lyndon words with trace 0 mod 6.

Original entry on oeis.org

1, 2, 11, 51, 259, 1282, 6665, 34938, 186612, 1007510, 5496925, 30231741, 167444795, 932900050, 5224277345, 29386526544, 165947641615, 940369781244, 5345260877285, 30467985992745, 174102782853475, 997134114520250
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054660, etc.

Extensions

More terms from James Sellers, Apr 19 2000

A054667 Number of 6-ary Lyndon words with trace 2 mod 6.

Original entry on oeis.org

1, 2, 12, 51, 259, 1284, 6665, 34938, 186624, 1007510, 5496925, 30231792, 167444795, 932900050, 5224277604, 29386526544, 165947641615, 940369782528, 5345260877285, 30467985992745, 174102782860140, 997134114520250
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 6-ary Lyndon words with trace 4 mod 6.

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054660, etc.

Extensions

More terms from James Sellers, Apr 19 2000

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Aug 11 2025

Keywords

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

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
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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