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.

A207969 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^4 * x^n/n ).

Original entry on oeis.org

1, 5, 15, 60, 295, 1625, 9430, 56465, 345010, 2139595, 13419500, 84926105, 541398665, 3472389210, 22385362895, 144945232375, 942089445030, 6143582084115, 40181143112035, 263482860974570, 1731780213622125, 11406235045261205, 75268685723935940
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k>=0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + 295*x^4 + 1625*x^5 + 9430*x^6 +...
such that
log(A(x))/5 = x + x^2/2 + 2^4*x^3/3 + 3^4*x^4/4 + 5^4*x^5/5 + 8^4*x^6/6 + 13^4*x^7/7 +...+ Fibonacci(n)^4*x^n/n +...

Crossrefs

Cf. A054888, A207970, A207971, A207972, A207834, A207835.
A077916, A203804.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^4*x^k/k)+x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + ... is an algebraic function: A(x)^5 = (1 + 3*x + x^2)^4/( (1 - 7*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A203804. - Peter Bala, Apr 03 2014
a(n) ~ 2^(4/5) * 5^(1/10) * phi^(4*n) / (Gamma(1/5) * 3^(1/5) * n^(4/5)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A207970 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^6 * x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 1505, 21875, 319620, 4936985, 77358485, 1236083870, 19982821875, 326511608255, 5379199407890, 89249496596015, 1489580814490755, 24988546214618750, 421055477328447620, 7122346563647277860, 120891417096833214485, 2058225554792946621495
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k >= 0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 1505*x^4 + 21875*x^5 + 319620*x^6 + ...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^6*x^4/4 + 5^6*x^5/5 + 8^6*x^6/6 + 13^6*x^7/7 + ... + Fibonacci(n)^6*x^n/n + ...

Crossrefs

Cf. A054888, A207969, A207971, A207972, A207834, A207834. A077916, A203806.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^6*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + ... is an algebraic function: A(x)^25 = ( (1 + 2*x + x^2)^10*(1 + 7*x + x^2)^6 )/( (1 - 3*x + x^2)^15*(1 - 18*x + x^2) ). Cf. A203806. - Peter Bala, Apr 03 2014
a(n) ~ 2^(17/25) * 5^(13/50) * phi^(6*n) / (Gamma(1/25) * 3^(3/5) * n^(24/25)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A211893 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^n * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 36, 561, 98211, 43176384, 116622937722, 1022189210900601, 41675008108242048327, 6377839090284322052067558, 4114890941608928235401688095580, 10460015732506081308723488849683574907, 108482611110966450613465001912856742180485969
Offset: 0

Views

Author

Paul D. Hanna, Apr 24 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 561*x^4 + 98211*x^5 + 43176384*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^3*x^3/3 + 5^4*x^4/4 + 11^5*x^5/5 + 21^6*x^6/6 + 43^7*x^7/7 +...+ Jacobsthal(n)^n*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Crossrefs

Cf. A211892, A211894, A211895, A211896, A207971, A207972, A001045.
Cf. A231292 (Jacobsthal(n)^n).

Programs

  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^k*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^n / 3^(n-1) * x^n/n ).

A207971 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*n) * x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 8795, 9808325, 57315191130, 2812698182891585, 894119494320160426760, 2048089587570930007354766745, 32079229816919862900907520464756250, 3500720882833094608324749707338857577696205, 2633228648869966875007549667526201212159637714889015
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Give g.f. A(x), note that A(x)^(1/5) is not an integer series.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 8795*x^4 + 9808325*x^5 +...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^8*x^4/4 + 5^10*x^5/5 + 8^12*x^6/6 + 13^14*x^7/7 +...+ Fibonacci(n)^(2*n)*x^n/n +...

Crossrefs

Cf. A054888, A207969, A207970, A207972, A207834, A207834.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^(2*k)*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 12, 198, 16962, 6762210, 11473594848, 80455865485692, 2306084412391039038, 268657100633050977422322, 126765866001055606588876061400, 241678197713843578271875740922972788, 1858396158245858742065123341776166504084452
Offset: 0

Views

Author

Paul D. Hanna, Apr 24 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 198*x^3 + 16962*x^4 + 6762210*x^5 +...
such that
log(A(x))/3 = x + 5*x^2/2 + 171*x^3/3 + 21845*x^4/4 + 11184811*x^5/5 + 22906492245*x^6/6 + 187649984473771*x^7/7 +...+ Jacobsthal(n^2)*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,...].

Crossrefs

Cf. A211893, A211894, A211895, A211896, A207972, A001045, A155200.
Cf. A231279 (Jacobsthal(n^2)).

Programs

  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

Formula

G.f.: (1+x) * exp( Sum_{n>=1} 2^(n^2) * x^n/n ).
a(n) = A155200(n) + A155200(n-1).
Showing 1-5 of 5 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.