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 1-8 of 8 results.

A202828 Expansion of e.g.f.: exp(4*x/(1-2*x)) / sqrt(1-4*x^2).

Original entry on oeis.org

1, 4, 36, 400, 5776, 97344, 1915456, 42406144, 1049760000, 28558296064, 848579961856, 27271456395264, 943132599095296, 34877026635366400, 1373536895379849216, 57351382681767706624, 2530646978003730497536, 117614221470591038521344, 5742190572014854792806400
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Comments

a(n) is the number of good involutions of the linear Alexander quandle (Z/4nZ, 2n+1); see Ta, Thm. 5.11 and cf. A387317. - Luc Ta, Aug 26 2025

Examples

			E.g.f.: A(x) = 1 + 4*x + 36*x^2/3! + 400*x^3/3! + 5776*x^4/4! + 97344*x^5/5! +...
where A(x) = 1 + 2^2*x + 6^2*x^2/2! + 20^2*x^3/3! + 76^2*x^4/4! + 312^2*x^5/5! +...+ A000898(n)^2*x^n/n! +...

Links

Crossrefs

Cf. A000898, A202827, A202829, A202831, A202833, A202835, A202836.
Cf. A387317.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022
  • Mathematica
    CoefficientList[Series[Exp[4*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
  • PARI
    {a(n)=n!*polcoeff(exp(4*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=sum(k=0,n\2,2^(n-2*k)*n!/((n-2*k)!*k!))^2}
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*2^k)^2}
  • SageMath
    def A202828_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202828_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A000898(n)^2, where the e.g.f. of A000898 is exp(2*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) = ( Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k) )^2. [From formula by Vladeta Jovovic in A000898].
a(n) ~ n^n*exp(2*sqrt(2*n)-1-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = 2*(n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013
a(n) = 2^n*A277378(n). - R. J. Mathar, Jan 20 2020

A202829 Expansion of e.g.f.: exp(4*x/(1-3*x)) / sqrt(1-9*x^2).

Original entry on oeis.org

1, 4, 49, 676, 13225, 293764, 7890481, 236359876, 8052729169, 300797402500, 12388985000401, 551925653637604, 26614517015830969, 1373655853915667716, 75803216516463190225, 4440662493517062816004, 275697752917311709134241, 18052104090118575573856516
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 49*x^2/2! + 676*x^3/3! + 13225*x^4/4! + 293764*x^5/5! + ...
were A(x) = 1 + 2^2*x + 7^2*x^2/2! + 26^2*x^3/3! + 115^2*x^4/4! + 542^2*x^5/5! + ... + A202830(n)^2*x^n/n! + ...

Links

Crossrefs

Cf. A202830, A202827, A202828, A202831, A202833, A202835, A202836.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-3*x))/Sqrt(1-9*x^2) ))); // G. C. Greubel, Jun 21 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-3x)]/Sqrt[1-9x^2],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Mar 09 2012 *)
  • PARI
    {a(n)=n!*polcoeff(exp(4*x/(1-3*x)+x*O(x^n))/sqrt(1-9*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))^2}
  • SageMath
    def A202829_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-3*x))/sqrt(1-9*x^2) ).egf_to_ogf().list()
A202829_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A202830(n)^2, where the e.g.f. of A202830 is exp(2*x + 3*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*3^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/3)-2/3-n)*3^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (3*n+1)*a(n-1) + 3*(n-1)*(3*n+1)*a(n-2) - 27*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202831 Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).

Original entry on oeis.org

1, 4, 81, 1444, 44521, 1397124, 58354321, 2574344644, 136043683281, 7657406908804, 489836445798001, 33351743794661604, 2504378700538997881, 199445618093659242244, 17189578072429077875121, 1564487078400498014277124, 152146464623361858013314721
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...

Links

Crossrefs

Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // G. C. Greubel, Jun 21 2022
  • Mathematica
    CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
  • PARI
    {a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}
  • PARI
    {a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}
  • SageMath
    def A202831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
A202831_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 4, 25, 196, 1849, 20164, 249001, 3422500, 51739249, 851822596, 15155825881, 289527934084, 5906625426025, 128089110981316, 2940882813228649, 71239270847432164, 1815115761586307041, 48511703775281296900, 1356708799439194070809, 39615996090901693902916
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 25*x^2/2! + 196*x^3/3! + 1849*x^4/4! +...
where A(x) = 1 + 2^2*x + 5^2*x^2/2! + 14^2*x^3/3! + 43^2*x^4/4! +...+ A005425(n)^2*x^n/n! +...

Links

Crossrefs

Cf. A005425, A202828, A202829, A202831, A202833, A202835, A202836.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 21 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-x)]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Dec 31 2011 *)
  • PARI
    {a(n)=n!*polcoeff(exp(4*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=sum(k=0,n\2,2^(n-3*k)*n!/((n-2*k)!*k!))^2}
  • SageMath
    def A202827_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202827_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A005425(n)^2, where the e.g.f. of A005425 is exp(2*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*n!/((n-2*k)!*k!) )^2. [From formula by Huajun Huang in A005425]
a(n) ~ n^n*exp(4*sqrt(n)-2-n)/2 * (1+5/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+3)*a(n-1) +(n-1)*(n+3)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 9, 100, 1296, 19044, 311364, 5588496, 108993600, 2291345424, 51585311376, 1236953249856, 31447331115264, 844332494760000, 23859653712215616, 707522071322329344, 21958125453144843264, 711555574637600891136, 24025060090437573945600
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 9*x + 100*x^2/2! + 1296*x^3/3! + 19044*x^4/4! + ...
where A(x) = 1 + 3^2*x + 10^2*x^2/2! + 36^2*x^3/3! + 138^2*x^4/4! + ... + A202834(n)^2*x^n/n! + ...

Links

Crossrefs

Cf. A202834, A202827, A202828, A202829, A202831, A202835, A202836.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(9*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[((9x)/(1-x))]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Apr 23 2012 *)
  • PARI
    {a(n)=n!*polcoeff(exp(9*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=n!^2*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)^2}
  • SageMath
    def A202833_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202833_list(40) # G. C. Greubel, Jun 22 2022

Formula

a(n) = A202834(n)^2, where the e.g.f. of A202834 is exp(3*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(6*sqrt(n)-9/2-n)/2 * (1+15/(4*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+8)*a(n-1) + (n-1)*(n+8)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202835 Expansion of e.g.f.: exp(9*x/(1-2*x)) / sqrt(1-4*x^2).

Original entry on oeis.org

1, 9, 121, 2025, 40401, 927369, 24000201, 689220009, 21710549025, 743187098889, 27441452694681, 1086166287819369, 45846179189949681, 2054407698719865225, 97357866191666622441, 4862830945258077841449, 255239441235423753980481, 14040944744510973314880009
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 9*x + 121*x^2/2! + 2025*x^3/3! + 40401*x^4/4! +...
where A(x) = 1 + 3^2*x + 11^2*x^2/2! + 45^2*x^3/3! + 201^2*x^4/4! + 963^2*x^5/5! +...+ A083886(n)^2*x^n/n! +...

Links

Crossrefs

Cf. A083886, A202827, A202828, A202829, A202831, A202833, A202836.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022
  • Mathematica
    CoefficientList[Series[Exp[9*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
  • PARI
    {a(n)=n!*polcoeff(exp(9*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=n!^2*polcoeff(exp(3*x+x^2+x*O(x^n)),n)^2}
  • PARI
    {a(n)=sum(k=0,n\2,3^(n-2*k)*n!/((n-2*k)!*k!))^2}
  • SageMath
    def A202835_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202835_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A083886(n)^2, where the e.g.f. of A083886 is exp(3*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(3*sqrt(2*n)-9/4-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (2*n+7)*a(n-1) + 2*(n-1)*(2*n+7)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202837 E.g.f.: exp(3*x + 2*x^2).

Original entry on oeis.org

1, 3, 13, 63, 345, 2043, 13029, 88119, 629169, 4707315, 36772029, 298608687, 2513795337, 21874602987, 196341166485, 1814001266727, 17222473789281, 167763502438371, 1674418724986221, 17102228350521375, 178562508150516921, 1903865792493260763
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: 1 + 3*x + 13*x^2/2! + 63*x^3/3! + 345*x^4/4! + 2043*x^5/5! +...

Links

Crossrefs

Cf. A202836.

Programs

  • Mathematica
    CoefficientList[Series[E^(3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    {a(n)=n!*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))}
  • PARI
    /* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+3*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-3*x-4*(n-k)*x^2*CF)); polcoeff(CF, n)}

Formula

a(n) = Sum_{k=0..[n/2]} 3^(n-2*k)*2^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-3*x - 4*x^2/(1-3*x - 8*x^2/(1-3*x - 12*x^2/(1-3*x - 16*x^2/(1-3*x -...))))), a continued fraction.
Recurrence: a(n) = 3*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(n-1/2)*exp(3/2*sqrt(n)-n/2-9/16)*n^(n/2)*(1+33/(64*sqrt(n))). - Vaclav Kotesovec, Oct 20 2012

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 1-8 of 8 results.

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