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 1-4 of 4 results.

A123916 Number of binary words whose (unique) decreasing Lyndon decomposition is into Lyndon words each with an odd number of 1's; EULER transform of A000048.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 34, 65, 120, 229, 432, 829, 1583, 3051, 5874, 11370, 22012, 42756, 83113, 161917, 315723, 616588, 1205232, 2358604, 4619485, 9055960, 17766086, 34880215, 68524486, 134707150, 264960828, 521449025
Offset: 1

Views

Author

Mike Zabrocki, Oct 28 2006

Keywords

Examples

			The binary words 1111, 1101, 1001, 0101, 0111, 0001 of length 4 decompose as 1*1*1*1, 1*1*01, 1*001, 01*01, 0111, 0001 and each subword has an odd number of 1's, therefore a(4)=6.
G.f. A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 19*x^7 + 34*x^8 + ... such that A(x)^2 * (1 - 2*x) = A(x^2).

Links

Crossrefs

Cf. A000048, A271929.

Programs

  • PARI
    /* G.f. A(x) satisfies: A(x)^2 = A(x^2)/(1 - 2*x) */
{a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x, x^2)/(1 - 2*x +x*O(x^n)))); polcoeff(A,n)}
for(n=1,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 17 2016
  • PARI
    /* As the EULER transform of A000048 */
{A000048(n) = sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n)} \\ Michael B. Porter
{a(n) = polcoeff( prod(k=1,n, 1/(1 - x^k +x*O(x^n))^A000048(k)), n-1)}
for(n=1,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 17 2016

Formula

Prod_{n>=1} 1/(1-q^n)^A000048(n) = 1 + sum_{n>=1} a(n) q^n.
G.f. A(x) satisfies: A(x)^2 = A(x^2) / (1 - 2*x). - Paul D. Hanna, Apr 17 2016
a(n) ~ c * 2^n / sqrt(n), where c = 0.3412831644583761326654... . - Vaclav Kotesovec, Apr 18 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (v^2 - 2*u^2*v - u^4) + 2*w*u^4. - Michael Somos, Jun 27 2017

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.

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 1-4 of 4 results.

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