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 1-2 of 2 results.

A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).

Original entry on oeis.org

1, 9, 169, 3969, 119025, 4173849, 169754841, 7764958161, 395853630561, 22158814509225, 1352182116776841, 89167147951863969, 6319166996322943569, 478498255838869322169, 38549853656690487255225, 3290600595687160597292529, 296613603422471046790496961
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 9*x + 169*x^2/2! + 3969*x^3/3! + 119025*x^4/4! + ...
where A(x) = 1 + 3^2*x + 13^2*x^2/2! + 63^2*x^3/3! + 345^2*x^4/4! + 2043^2*x^5/5! + ... + A202837(n)^2*x^n/n! + ...

Links

Crossrefs

Cf. A202837, A202827, A202828, A202829, A202831, A202833, A202835.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-4*x))/Sqrt(1-16*x^2) ))); // G. C. Greubel, Jun 22 2022
  • Mathematica
    CoefficientList[Series[Exp[9*x/(1-4*x)]/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
  • PARI
    {a(n)=n!*polcoeff(exp(9*x/(1-4*x)+x*O(x^n))/sqrt(1-16*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=n!^2*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)^2}
  • PARI
    {a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))^2}
  • SageMath
    def A202836_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-4*x))/sqrt(1-16*x^2) ).egf_to_ogf().list()
A202836_list(40) # G. C. Greubel, Jun 22 2022

Formula

a(n) = A202837(n)^2, where the e.g.f. of A202837 is exp(3*x + 2*x^2).
a(n) ~ n^n*exp(3*sqrt(n)-9/8-n)*2^(2*n-1) * (1+33/(32*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (4*n+5)*a(n-1) + 4*(n-1)*(4*n+5)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202878 Expansion of e.g.f.: exp(16*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 16, 289, 5776, 126025, 2972176, 75186241, 2027520784, 57988974481, 1751546371600, 55668326576641, 1855807478279056, 64713593898036889, 2354701531657512976, 89209297718289390625, 3512141211682081889296, 143435878498076017059361
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 16*x + 289*x^2/2! + 5776*x^3/3! + 126025*x^4/4! + ...
where A(x) = 1 + 4^2*x + 17^2*x^2/2! + 76^2*x^3/3! + 355^2*x^4/4! + 1724^2*x^5/5! + ... + A202879(n)^2*x^n/n! + ...

Links

Crossrefs

Cf. A202879, A202827, A202828, A202829, A202831, A202835, A202836, A202837.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(16*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022
  • Mathematica
    CoefficientList[Series[Exp[16*x/(1-x)]/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
  • PARI
    {a(n)=n!*polcoeff(exp(16*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=n!^2*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)^2}
  • SageMath
    def A202878_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(16*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202878_list(40) # G. C. Greubel, Jun 22 2022

Formula

a(n) = A202879(n)^2, where the e.g.f. of A202879 is exp(4*x + x^2/2).
a(n) = ( Sum_{k=0..floor(n/2)} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(8*sqrt(n)-8-n)/2 * (1+22/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+15)*a(n-1) + (n-1)*(n+15)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013
Showing 1-2 of 2 results.

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