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.

Showing 1-6 of 6 results.

A110540 Invertible triangle: T(n,k) = number of k-ary Lyndon words of length n-k+1 with trace 1 modulo k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 3, 2, 1, 0, 3, 6, 5, 2, 1, 0, 5, 16, 16, 8, 3, 1, 0, 9, 39, 51, 30, 12, 3, 1, 0, 16, 104, 170, 125, 54, 16, 4, 1, 0, 28, 270, 585, 516, 259, 84, 21, 4, 1, 0, 51, 729, 2048, 2232, 1296, 480, 128, 27, 5, 1, 0, 93, 1960, 7280, 9750, 6665, 2792, 819, 180, 33, 5, 1
Offset: 1

Views

Author

Paul Barry, Jul 25 2005

Keywords

Comments

An invertible number triangle related to Lyndon words of trace 1.

Examples

			Rows begin
  1;
  0,  1;
  0,  1,   1;
  0,  1,   1,    1;
  0,  2,   3,    2,    1;
  0,  3,   6,    5,    2,    1;
  0,  5,  16,   16,    8,    3,   1;
  0,  9,  39,   51,   30,   12,   3,   1;
  0, 16, 104,  170,  125,   54,  16,   4,  1;
  0, 28, 270,  585,  516,  259,  84,  21,  4, 1;
  0, 51, 729, 2048, 2232, 1296, 480, 128, 27, 5, 1;

Links

Crossrefs

Columns include A000048, A046211, A054660, A054662, A054666, A373277, A300674, A300675.

Programs

  • Mathematica
    T[n_, k_]:=Sum[Boole[GCD[d, k] == 1]  MoebiusMu[d] k^((n - k + 1)/d), {d, Divisors[n - k + 1]}] /(k(n - k + 1)); Flatten[Table[T[n, k], {n, 12}, {k, n}]] (* Indranil Ghosh, Mar 27 2017 *)
  • PARI
    for(n=1, 11, for(k=1, n, print1( sum(d=1,n-k+1, if(Mod(n-k+1, d)==0 && gcd(d, k)==1, moebius(d)*k^((n-k+1)/d), 0)/(k*(n-k+1)) ),", ");); print();) \\ Andrew Howroyd, Mar 26 2017

Formula

T(n, k) = (Sum_{d | n-k+1, gcd(d, k)=1} mu(d)*k^((n-k+1)/d))/(k*(n-k+1)).

Extensions

Name clarified by Andrew Howroyd, Mar 26 2017

A373281 Expansion of Sum_{k>=0} x^(5^k) / (1 - 5*x^(5^k)).

Original entry on oeis.org

1, 5, 25, 125, 626, 3125, 15625, 78125, 390625, 1953130, 9765625, 48828125, 244140625, 1220703125, 6103515650, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328250, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125
Offset: 1

Views

Author

Seiichi Manyama, May 30 2024

Keywords

Links

Crossrefs

Cf. A187767, A373279, A373280, A373282, A373283.
Cf. A000351, A054662, A055457.

Programs

  • PARI
    b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=5) = sumdiv(n, d, d*b(d, k));

Formula

G.f. A(x) satisfies A(x) = x/(1 - 5*x) + A(x^5).
If n == 0 (mod 5), a(n) = 5^n + a(n/5) otherwise a(n) = 5^n.
a(n) = Sum_{d|n} d * A054662(d).

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

A372956 G.f. A(x) satisfies A(x)^5 = A(x^5) / (1 - 5*x)^5 with A(0)=1.

Original entry on oeis.org

1, 5, 25, 125, 625, 3126, 15630, 78150, 390750, 1953750, 9768753, 48843765, 244218825, 1221094125, 6105470625, 30527353136, 152636765680, 763183828400, 3815919142000, 19079595710000, 95397978550044, 476989892750220, 2384949463751100, 11924747318755500, 59623736593777500
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2024

Keywords

Comments

Euler transform of 5 * A054662(n).

Examples

			A(x)^5 = 1 + 25*x + 375*x^2 + 4375*x^3 + 43750*x^4 + 393755*x^5 + ... .

Links

Crossrefs

Cf. A372870, A372957.
Cf. A054662, A372535.

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, 5))^5)

Formula

G.f.: A(x) = 1 / ( Product_{k>=1} (1 - x^k)^A054662(k) )^5.

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

A074026 Duplicate of A054663.

Original entry on oeis.org

1, 2, 8, 30, 124, 516, 2232, 9750, 43400, 195248, 887784, 4068740, 18780048, 87191964
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 21 2002

Keywords

Comments

Also number of 5-ary Lyndon words of trace 0 over GF(5).

Links

Crossrefs

Cf. A054662.
Showing 1-6 of 6 results.

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