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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.

User: Sergei N. Gladkovskii

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Sergei N. Gladkovskii has authored 2 sequences.

A258723 Expansion of 1/(1-12*x+48*x^2)^(1/2).

Original entry on oeis.org

1, 6, 30, 108, 54, -3564, -41364, -314280, -1798362, -6972156, -1793340, 283697640, 3341429820, 25984971720, 151750943640, 596184213168, 101849014278, -25747257110940, -305001821608236, -2392882855430328, -14088646343199276, -55649498057805096, -7100681134947480
Offset: 0

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Author

Sergei N. Gladkovskii, Jun 08 2015

Keywords

Links

Crossrefs

Cf. A144535, A144536, A240880.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-12*x+48*x^2)^(1/2), {x, 0, 20}], x]  (* Vaclav Kotesovec, Jun 08 2015 *)
  • PARI
    Vec(1/(1-12*x+48*x^2)^(1/2) + x^50) \\ G. C. Greubel, Feb 14 2017

Formula

G.f.: 1/(1-12*x+48*x^2)^(1/2).
E.g.f.: exp(6*x)*BesselJ(0,2*sqrt(3)*x).
If mod(n,6)=4 then a(n) ~ (-1)^((n+8)/6) * 3^((n+1)/2) * 4^(n-1) / (sqrt(Pi) * n^(3/2)), else a(n) ~ 3^(n/2) * 2^(2*n+1) * cos(Pi*(n-1)/6) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 08 2015
D-finite with recurrence n*a(n) +6*(-2*n+1)*a(n-1) +48*(n-1)*a(n-2)=0. [Belbachir]

A201158 E.g.f. exp(x)/(cos(x) - sin(x)).

Original entry on oeis.org

1, 2, 6, 24, 124, 792, 6056, 53984, 549904, 6301472, 80233056, 1123714944, 17169102784, 284184941952, 5065697161856, 96747688891904, 1970927736619264, 42660873261343232, 977715195437139456, 23652447354912036864, 602304626050881977344
Offset: 0

Views

Author

Sergei N. Gladkovskii, Nov 27 2011

Keywords

Comments

Sum_{n>=0} a(n)x^n/n! = exp(x)/(cos(x) - sin(x)).

Programs

  • Maple
    A:=series(exp(x)/(cos(x)-sin(x)),x,40);
G(x):=A : f[0]:=G(x): for n from 1 to 41 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..40);
  • Mathematica
    nn = 30; Range[0, nn]! CoefficientList[Series[Exp[x]/(Cos[x] - Sin[x]), {x, 0, nn}], x] (* T. D. Noe, Dec 05 2011 *)

Formula

Sum_{n>=0} a(n)x^n/n! = exp(x)/(cos(x) - sin(x)).
exp(x)/(cos(x) - sin(x)) = 1/G(0); G(k) = 1-2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1)-x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction).
G.f.: 1/G(0) where G(k) = 1 - 2*x*(k+1) - 2*x^2*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
a(n) ~ n! * exp(Pi/4)*2^(2*n+3/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 27 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 2*x*k)*(1 - 4*x - 2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013

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