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 1-4 of 4 results.

A007289 Expansion of e.g.f. (sin(2*x) + cos(x)) / cos(3*x).

Original entry on oeis.org

1, 2, 8, 46, 352, 3362, 38528, 515086, 7869952, 135274562, 2583554048, 54276473326, 1243925143552, 30884386347362, 825787662368768, 23657073914466766, 722906928498737152, 23471059057478981762, 806875574817679474688, 29279357851856595135406
Offset: 0

Views

Author

N. J. A. Sloane and Simon Plouffe

Keywords

Comments

Arises in the enumeration of alternating 3-signed permutations.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row 3 of A349271.
Cf. A006873, A007286, A225109, A000191 (bisection), A000436 (bisection).
Cf. A000111, A136630, A352638.

Programs

  • Maple
    A007289 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k),j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011
  • Mathematica
    mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[2 x] + Cos[x])/Cos[3 x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace((sin(2*x) + cos(x)) / cos(3*x))) \\ Joerg Arndt, Apr 28 2013
  • Sage
    from mpmath import mp, polylog, im
mp.dps = 32; mp.pretty = True
def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
def A007289(n) : return im(aperm3(n))
[int(A007289(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

Formula

E.g.f.: (sin(2*x) + cos(x)) / cos(3*x).
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k). - Peter Luschny, Jul 31 2011
a(n) = Im(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - Peter Luschny, Apr 28 2013
a(n) ~ n! * 2^(n+1)*3^(n+1/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
From Seiichi Manyama, Jun 25 2025: (Start)
E.g.f.: 1/(1 - 2 * sin(x)).
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)

A000813 Expansion of (sin x + cos x)/cos 4x.

Original entry on oeis.org

1, 1, 15, 47, 1185, 6241, 230895, 1704527, 83860545, 796079041, 48942778575, 567864586607, 41893214676705, 574448847467041, 49441928730798255, 782259922208550287, 76946148390480577665, 1379749466246228538241
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

a(2n) = A001728(n). Cf. A006873, A156201, A156205.

Programs

  • Maple
    p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A000813 := n -> -(-1)^iquo(n,2)*Re(p(n));
seq(A000813(i),i=0..11);  # Peter Luschny, Apr 29 2013
  • Mathematica
    a[n_] := 2*(-1)^Floor[n/2]*Im[Sum[4^j*Binomial[n, j]*PolyLog[-j, I], {j, 0, n}]]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 30 2013, after Peter Luschny *)
With[{nn=20},CoefficientList[Series[(Sin[x]+Cos[x])/Cos[4x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 12 2013 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace((sin(x)+cos(x))/cos(4*x))) \\ Joerg Arndt, Apr 30 2013

Formula

a(n) = -(-1)^floor(n/2)*Re(2*I*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*4^j))). - Peter Luschny, Apr 29 2013

A007286 E.g.f.: (sin x + cos 2x) / cos 3x.

Original entry on oeis.org

1, 1, 5, 26, 205, 1936, 22265, 297296, 4544185, 78098176, 1491632525, 31336418816, 718181418565, 17831101321216, 476768795646785, 13658417358350336, 417370516232719345, 13551022195053101056
Offset: 0

Views

Author

N. J. A. Sloane and Simon Plouffe

Keywords

Comments

Arises in the enumeration of alternating 3-signed permutations.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006873, A007289, A225109, A002438 (bisection?).

Programs

  • Mathematica
    mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[x] + Cos[2x])/Cos[3 x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace((sin(x)+cos(2*x))/cos(3*x))) \\ Joerg Arndt, Apr 28 2013
  • Sage
    from mpmath import mp, polylog, re
mp.dps = 32; mp.pretty = True
def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
def A007286(n) : return re(aperm3(n))
[int(A007286(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

Formula

a(n) = Re(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - Peter Luschny, Apr 28 2013
a(n) ~ n! * 2^(n+1)*3^n/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013

A225109 E.g.f. (sin(3x) + cos x) / cos(4x).

Original entry on oeis.org

1, 3, 15, 117, 1185, 15123, 230895, 4116837, 83860545, 1921996323, 48942778575, 1370953667157, 41893214676705, 1386843017916723, 49441928730798255, 1888542637550347077, 76946148390480577665, 3331009898404800736323, 152682246738275154625935, 7387240827905368219116597
Offset: 0

Views

Author

M. F. Hasler, Apr 28 2013

Keywords

Links

Crossrefs

Cf. A006873, A007289, A007286.

Programs

  • Maple
    per4 := proc(n) local j; 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)* polylog(-j,I)*4^j, j=0..n)) end: A225109 := n -> Im(per4(n));
seq(A225109(i), i=0..11); # Peter Luschny, Apr 29 2013
  • Mathematica
    mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[3x] + Cos[x])/Cos[4x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)
  • PARI
    v=Vec((sin(3*x) + cos(x)) / cos(4*x)); vector(#v,i,v[i]*(i-1)!)
  • PARI
    x='x+O('x^66); Vec(serlaplace((sin(3*x)+cos(x))/cos(4*x))) \\ Joerg Arndt, Apr 28 2013

Formula

a(n) = Im(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)* 4^j))). - Peter Luschny, Apr 29 2013
a(n) ~ n! * sqrt(2+sqrt(2)) * 2^(3*n+1)/Pi^(n+1). - Vaclav Kotesovec, Jun 02 2013
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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