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 1-5 of 5 results.

A271929 G.f. A(x) satisfies: A(x)^3 = A(x^3) / (1 - 3*x).

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 83, 224, 615, 1708, 4777, 13455, 38110, 108428, 309714, 887666, 2551575, 7353423, 21240460, 61478489, 178269670, 517784717, 1506162369, 4387201004, 12795170784, 37359689295, 109199349181, 319493390481, 935616592227, 2742209152877, 8043500169958, 23610710680582, 69354125493930, 203852682699869, 599549063015417, 1764338532368820
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2016

Keywords

Comments

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

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 31*x^6 + 83*x^7 + 224*x^8 + 615*x^9 + 1708*x^10 + 4777*x^11 + 13455*x^12 +...
where A(x)^3 = A(x^3) / (1 - 3*x).
Also, when expressed as the EULER transform of A046211,
A(x) = x/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^6 * (1-x^5)^16 * (1-x^6)^39 * (1-x^7)^104 * (1-x^8)^270 * (1-x^9)^729 *...* (1-x^n)^A046211(n) *...).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 84*x^7 + 252*x^8 + 758*x^9 + 2274*x^10 + 6822*x^11 + 20471*x^12 + 61413*x^13 + 184239*x^14 +...

Links

Crossrefs

Cf. A123916.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3)/(1 - 3*x +x*O(x^n)))^(1/3)); polcoeff(G=A, n)}
for(n=1, 50, print1(a(n), ", "))

Formula

The EULER transform of A046211, where A046211(n) is the number of ternary Lyndon words whose digits sum to 1 (or 2) mod 3.
a(n) ~ c * 3^n / n^(2/3), where c = 0.1260671867244258410294918... . - Vaclav Kotesovec, Apr 18 2016

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

A372957 G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 - 2*x)^2 with A(0)=1.

Original entry on oeis.org

1, 2, 5, 10, 22, 44, 91, 182, 370, 740, 1490, 2980, 5979, 11958, 23950, 47900, 95865, 191730, 383580, 767160, 1534549, 3069098, 6138628, 12277256, 24555341, 49110682, 98222947, 196445894, 392894839, 785789678, 1571585230, 3143170460, 6286352290, 12572704580, 25145431172
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2024

Keywords

Comments

Euler transform of 2 * A000048(n).

Examples

			A(x)^2 = 1 + 4*x + 14*x^2 + 40*x^3 + 109*x^4 + 276*x^5 + 678*x^6 + ... .

Links

Crossrefs

Cf. A372870, A372956.
Cf. A000048, A123916.

Programs

  • PARI
    b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1 - x^k)^b(k, 2))^2)

Formula

G.f.: A(x) = 1 / ( Product_{k>=1} (1 - x^k)^A000048(k) )^2.

A329276 Expansion of 1 / (1 - Sum_{k>=1} mu(2*k) * log(1 - 2 * x^k) / (2 * k)), where mu = A008683.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 102, 232, 528, 1204, 2748, 6276, 14342, 32787, 74976, 171495, 392337, 897696, 2054232, 4701202, 10759689, 24627245, 56370546, 129034271, 295373313, 676158166, 1547869038, 3543458906, 8111974160, 18570800837, 42514665175, 97330789942, 222825306335
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 11 2019

Keywords

Comments

Invert transform of A000048.

Crossrefs

Cf. A000048, A008683, A123916, A329275.

Programs

  • Mathematica
    nmax = 33; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k] Log[1 - 2 x^k]/(2 k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(1/(2 k)) DivisorSum[k, MoebiusMu[#] 2^(k/#) &, OddQ] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000048(k) * a(n-k).

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
Showing 1-5 of 5 results.

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