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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A204062 Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.

Original entry on oeis.org

1, 4, 26, 148, 867, 5048, 29428, 171512, 999653, 5826396, 33958734, 197925996, 1153597255, 6723657520, 39188347880, 228406429744, 1331250230601, 7759094953844, 45223319492482, 263580822001028, 1536261612513707, 8953988853081192, 52187671505973468
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...

Links

Crossrefs

Cf. A000129, A002203, A026933, A204061, A212442.

Programs

  • Magma
    I:=[1,4,26,148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 25 2021
  • Mathematica
    LinearRecurrence[{4,10,4,-1},{1,4,26,148},30] (* Vincenzo Librandi, Feb 12 2012 *)
Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* G. C. Greubel, May 25 2021 *)
  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}
  • Sage
    [(lucas_number1(2*n+4,2,-1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # G. C. Greubel, May 25 2021

Formula

G.f.: 1/((1+x)^2*(1-6*x+x^2)).
Self-convolution of A026933.
Self-convolution 4th power of A204061.
a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).
a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - Bruno Berselli, Jan 10 2012
a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - G. C. Greubel, May 25 2021

A212443 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.

Original entry on oeis.org

4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, 4523232, 23970240, 128063040, 689008320, 3728973120, 20285199872, 110841302880, 608029121280, 3346972244000, 18480871268160, 102328688556864, 568014587806720, 3160148362953120, 17617881702072960
Offset: 1

Views

Author

Paul D. Hanna, May 17 2012

Keywords

Crossrefs

Cf. A008683, A212442, A203853, A002203.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * LucasL[#, 2]^2 &] / n; Array[a, 25] (* Amiram Eldar, Aug 22 2023 *)
  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*A002203(d)^2)/n)}
for(n=1,30,print1(a(n),","))

Formula

G.f.: 1/Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} A002203(n)^3 * x^n/n), which equals the g.f. of A212442.

A215171 G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 16, 776, 23856, 834596, 28135056, 957599096, 32515276336, 1104679254346, 37525681919856, 1274775209167896, 43304782313176656, 1471088177488196276, 49973690736096892016, 1697634414511896630376, 57669596280038205388752, 1959068639950002397935907
Offset: 0

Views

Author

Paul D. Hanna, Aug 05 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A002203(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A002203(2*k)*x - x^2)^binomial(2*n,n-k).
Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

Examples

			G.f.: A(x) = 1 + 16*x + 776*x^2 + 23856*x^3 + 834596*x^4 + 28135056*x^5 +...
where
log(A(x)) = 2^4*x + 6^4*x^2/2 + 14^4*x^3/3 + 34^4*x^4/4 + 82^4*x^5/5 + 198^4*x^6/6 + 478^4*x^7/7 + 1154^4*x^8/8 +...+ A002203(n)^4*x^n/n +...

Crossrefs

Cf. A204062, A212442, A203804.

Programs

  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^4*x^k/k)+x*O(x^n)), n)}
  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{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)*A002203(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}

Formula

G.f.: 1/((1-x)^6*(1+6*x+x^2)^4*(1-34*x+x^2)).
Showing 1-3 of 3 results.

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