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.

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A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

Original entry on oeis.org

1, 1, 5, 85, 2928, 314925, 84974760, 63327890015, 123670531939440, 644385861467631972, 8853970669063185618000, 321538767413685546538468385, 30768712746239178236068160093280, 7755868453482819803691622493685140880, 5144106193113274410507722020733942141881664
Offset: 1

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^85 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^314925 * (1-29*x^7-x^14)^84974760 * (1-47*x^8+x^16)^63327890015 * (1-76*x^9-x^18)^123670531939440 *...).
where F(x) = exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = g.f. of A156216:
F(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
so that the logarithm of F(x) begins:
log(F(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 + 29^7*x^7/7 + 47^8*x^8/8 + 76^9*x^9/9 + 123^10*x^10/10 +...+ Lucas(n)^n*x^n +...

Links

Crossrefs

Cf. A156216, A000032 (Lucas), A203318, A203803, A203804, A203805, A203806, A203807, A203808.

Programs

  • Mathematica
    a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^(#-1)&]; Array[a, 15] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^(d-1))/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^m*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n), which is the g.f. of A156216.
G.f.: Product_{n>=1} G_n(x^n)^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A203803 G.f.: exp( Sum_{n>=1} A000204(n)^3 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 14, 35, 205, 744, 3414, 13926, 60060, 252330, 1072902, 4537272, 19234463, 81452015, 345084970, 1461714517, 6192083147, 26229794928, 111111714300, 470675847900, 1993816532280, 8445939457380, 35777578796220, 151556246864400, 642002579853325, 2719566542567917
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Examples

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 +...
where
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 +...+ Lucas(n)^3*x^n/n +...

Links

Crossrefs

Cf. A002571, A203804, A203805, A203806, A203807, A203808, A203809.
Cf. A203853, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 + x - x^2)^3*(1 - 4*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^3*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=1)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

Formula

G.f.: 1/( (1+x-x^2)^3 * (1-4*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203853(n) where A203853(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^2.

A203804 G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 41, 126, 1526, 7854, 63629, 400789, 2870629, 19254504, 133376760, 909578760, 6249172910, 42785312510, 293403088510, 2010553849020, 13781960765020, 94458627485820, 647442212896270, 4437595353800270, 30415849505902910, 208472981440853160, 1428896115173689560
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

Examples

			G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...

Links

Crossrefs

Cf. A002571, A203803, A203805, A203806, A203807, A203808, A203809.
Cf. A203854, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - x)^6*(1 + 3*x + x^2)^4*(1 - 7*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^4*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

Formula

G.f.: 1/( (1-x)^6 * (1+3*x+x^2)^4 * (1-7*x+x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203854(n) where A203854(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^3.

A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 365, 1730, 97390, 948562, 26292937, 370813165, 7716851405, 127699557640, 2397734250216, 42004273130216, 763345960355450, 13608990417046650, 245008471017094450, 4389301146029065420, 78826300825689660420, 1413927351334191841100, 25376664633745265522450
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

Examples

			G.f.: A(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
where
log(A(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...

Links

Crossrefs

Cf. A002571, A203803, A203804, A203805, A203807, A203808, A203809.
Cf. A203856, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 + x)^20*(1 - 3*x + x^2)^15*(1 + 7*x + x^2)^6*(1 - 18*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^6*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=3)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

Formula

G.f.: 1/( (1+x)^20 * (1-3*x+x^2)^15 * (1+7*x+x^2)^6 * (1-18*x+x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203856(n) where A203856(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^5.

A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 122, 463, 11985, 85456, 1262166, 12018742, 145326748, 1540766090, 17495016342, 191731126832, 2138972609189, 23652975370501, 262682339212290, 2911255335387883, 32296421465575573, 358120616523262016, 3971885483375619384, 44047530724737577400
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Examples

			G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
where
log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...

Links

Crossrefs

Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.
Cf. A203855, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=2)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

Formula

G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.

A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 3281, 25126, 6845526, 121368902, 12805025677, 373879862237, 24707348223677, 948781359159752, 50702478932197928, 2210812262034197128, 108528095366637700218, 4974402150387759436378, 236926456045384849970778, 11047772769135934828000404
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

Examples

			G.f.: A(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(A(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...

Links

Crossrefs

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203809.
Cf. A203858, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - x)^70*(1 + 3*x + x^2)^56*(1 - 7*x + x^2)^28*(1 + 18*x + x^2)^8*(1 - 47*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^8*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=4)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

Formula

G.f.: 1/( (1-x)^70 * (1+3*x+x^2)^56 * (1-7*x+x^2)^28 * (1+18*x+x^2)^8 * (1-47*x+x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203858(n) where A203858(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^7.

A203809 G.f.: exp( Sum_{n>=1} A000204(n)^9 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 9842, 97223, 58608265, 1390114224, 296390076414, 12122505505998, 1486321234837932, 84428445979241330, 7833461016478812734, 528228569507280147664, 43275470600883540869733, 3148637876123977595284117, 245565185017744596492591850
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Examples

			G.f.: A(x) = 1 + x + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 + ...
where
log(A(x)) = x + 3^9*x^2/2 + 4^9*x^3/3 + 7^9*x^4/4 + 11^9*x^5/5 + 18^9*x^6/6 + 29^9*x^7/7 + 47^9*x^8/8 + ... + Lucas(n)^9*x^n/n + ...

Links

Crossrefs

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203808.
Cf. A203859, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - x - x^2)^126*(1 + 4*x - x^2)^84*(1 - 11*x - x^2)^36*(1 + 29*x - x^2)^9*(1 - 76*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^9*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=4)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

Formula

G.f.: 1/( (1-x-x^2)^126 * (1+4*x-x^2)^84 * (1-11*x-x^2)^36 * (1+29*x-x^2)^9 * (1-76*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203859(n) where A203859(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^8.

A203857 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 364, 1365, 29230, 354312, 5667900, 84974760, 1347387210, 21411102720, 346282421940, 5645803690800, 92886793449030, 1538448587832240, 25635241395476100, 429333683845968552, 7222607529064709670, 121980435560782376760, 2067248664062116147200
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^364 * (1-4*x^3-x^6)^1365 * (1-7*x^4+x^8)^29230 * (1-11*x^5-x^10)^354312 * (1-18*x^6+x^12)^5667900 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^7 * x^n/n ) = g.f. of A203807:
F(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where log(F(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...

Links

Crossrefs

Cf. A203807, A203853, A203854, A203855, A203856, A203858, A203859, A203800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^6&]/n; Array[a, 18] (* Jean-François Alcover, Dec 07 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^6)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^7*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^7 * x^n/n), which is the g.f. of A203807.
a(n) ~ phi^(6*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017
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