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.

A380551 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).

Original entry on oeis.org

1, 1, 6, 28, 142, 720, 3875, 21288, 120168, 690546, 4032014, 23840724, 142498691, 859512043, 5225263875, 31983651216, 196947587822, 1219199232294, 7583142491924, 47365473951152, 296983176365613, 1868545308601424, 11793499763070479, 74650344221104632, 473770694965305205, 3014124873709172435
Offset: 1

Views

Author

Paul D. Hanna, Feb 16 2025

Keywords

Comments

Moebius transform of A006013.

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 28*x^4 + 142*x^5 + 720*x^6 + 3875*x^7 + 21288*x^8 + 120168*x^9 + 690546*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + ... + A006013(n)*x^(n+1) + ...
which equals x*F(x)^2 where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

Links

Crossrefs

Cf. A346925, A034742, A380552, A380553, A006013, A001764, A008683.

Programs

  • PARI
    \\ As the Moebius transform of A006013 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(3*d-1,d-1)*2/(3*d-1) )}
for(n=1,30,print1(a(n),", "))
  • PARI
    \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(2*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(2*n) / (1 - x^n*(1-x)^(2*n)).
(3) x*F(x)^2 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(3*d-1,d-1)*2/(3*d-1), where mu is the Moebius function A008683.

A380552 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).

Original entry on oeis.org

1, 2, 14, 88, 611, 4372, 32889, 254384, 2017341, 16300550, 133767542, 1111727456, 9338434699, 79155402978, 676196048434, 5815796615520, 50318860986107, 437662918037250, 3824609516638443, 33563127916092808, 295655735395364616, 2613391671434553220, 23173063762591336049, 206066197523415007168
Offset: 1

Views

Author

Paul D. Hanna, Feb 16 2025

Keywords

Comments

Moebius transform of A006632.

Examples

			G.f.: A(x) = x + 2*x^2 + 14*x^3 + 88*x^4 + 611*x^5 + 4372*x^6 + 32889*x^7 + 254384*x^8 + 2017341*x^9 + 16300550*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 3*x^2 + 15*x^3 + 91*x^4 + 612*x^5 + 4389*x^6 + 32890*x^7 + 254475*x^8 + ... + A006632(n)*x^(n) + ...
which equals x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

Links

Crossrefs

Cf. A346935, A034742, A380551, A380553, A006632, A002293, A008683.

Programs

  • PARI
    \\ As the Moebius transform of A006632 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(4*d-1,d-1)*3/(4*d-1) )}
for(n=1,30,print1(a(n),", "))
  • PARI
    \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(3*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(3*n) / (1 - x^n*(1-x)^(3*n)).
(3) x*F(x)^3 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(4*d-1,d-1)*3/(4*d-1), where mu is the Moebius function A008683.

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

Original entry on oeis.org

1, 3, 25, 200, 1770, 16351, 158223, 1577328, 16112031, 167708890, 1772645419, 18974340640, 205263418940, 2240623110285, 24648785800540, 272994642782048, 3041495503591364, 34064252952038769, 383302465665133013, 4331178750570145160, 49126274119206904221, 559128033687856289017
Offset: 1

Views

Author

Paul D. Hanna, Feb 16 2025

Keywords

Comments

Moebius transform of A118971.

Examples

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 200*x^4 + 1770*x^5 + 16351*x^6 + 158223*x^7 + 1577328*x^8 + 16112031*x^9 + 167708890*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 4*x^2 + 26*x^3 + 204*x^4 + 1771*x^5 + 16380*x^6 + 158224*x^7 + 1577532*x^8 + ... + A118971(n)*x^(n) + ...
which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

Links

Crossrefs

Cf. A346936, A034742, A380551, A380552, A118971, A002294, A008683.

Programs

  • PARI
    \\ As the Moebius transform of A118971 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
for(n=1,30,print1(a(n),", "))
  • PARI
    \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(4*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(4*n) / (1 - x^n*(1-x)^(4*n)).
(3) x*F(x)^4 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(5*d-1,d-1)*4/(5*d-1), where mu is the Moebius function A008683.

A251662 Dirichlet convolution of Moebius function mu(n) (A008683) with Ternary numbers A001764.

Original entry on oeis.org

1, 0, 2, 11, 54, 270, 1427, 7740, 43260, 246620, 1430714, 8414356, 50067107, 300829144, 1822766463, 11124747912, 68328754958, 422030501802, 2619631042664, 16332922043614, 102240109896265, 642312449787030, 4048514844039119, 25594403732709300, 162250238001816845, 1031147983109715120
Offset: 1

Views

Author

Paul D. Hanna, Jan 04 2015

Keywords

Examples

			G.f.: A(x) = x + 2*x^3 + 11*x^4 + 54*x^5 + 270*x^6 + 1427*x^7 + 7740*x^8 +...
where Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x - x^2:
x-x^2 = A(x*(1-x)^2) + A(x^2*(1-x)^4) + A(x^3*(1-x)^6) + A(x^4*(1-x)^8) +...

Crossrefs

Cf. A034742, A001764, A008683.

Programs

  • PARI
    /* Dirichlet convolution of mu(n) with Ternary numbers A001764: */
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1))}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    /* G.f. satisfies: Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x-x^2. */
{a(n)=local(A=[1, 0]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(sum(n=1, #A, subst(x*Ser(A), x, (x-2*x^2+x^3 +x*O(x^#A))^n)))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: Sum_{n>=1} A((x - 2*x^2 + x^3)^n) = x - x^2.
a(n) = Sum_{d|n} Moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.