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 1-6 of 6 results.

A285043 Expansion of cosh(3*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 18, 102, 500, 2310, 10332, 45276, 195624, 836550, 3549260, 14965236, 62783448, 262303132, 1092063000, 4533175800, 18769219920, 77539370310, 319704052140, 1315894618500, 5407825361400, 22193291140020
Offset: 0

Views

Author

Peter Bala, Apr 09 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4. For n = 0 we get the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285044, A285045, A285046.

Programs

  • Maple
    seq((8*n + 1)*binomial(2*n,n), n = 0..20);
  • Mathematica
    CoefficientList[Series[Cosh[3*ArcTanh[2*Sqrt[x]]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 10 2017 *)
  • PARI
    my(x='x + O('x^30)); Vec((1 + 12*x)/(1 - 4*x)^(3/2)) \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = (8*n + 1)*binomial(2*n,n).
O.g.f. cosh(3*arctanh(2*sqrt(x))) = (1 + 12*x)/(1 - 4*x)^(3/2) = 1 + 18*x + 102*x^2 + 500*x^3 + ....
D-finite with recurrence: n*a(n) +2*(4*n-13)*a(n-1) +24*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

A285046 Expansion of cosh(9*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 162, 4806, 71892, 758214, 6506172, 48783900, 332715240, 2115552582, 12745645484, 73577414196, 410265444888, 2222886926364, 11756568121560, 60911288332920, 310024235290320, 1553692427724870
Offset: 0

Views

Author

Peter Bala, Apr 10 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n + 1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285044, A285045.

Programs

  • Maple
    seq(1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n), n = 0..20);
  • PARI
    x='x + O('x^30); print(Vec((1 + 144*x + 2016*x^2 + 5376*x^3 + 2304*x^4)/(1 - 4*x)^(9/2))) \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = 1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n).
O.g.f. cosh(9*arctanh(2*sqrt(x))) = (1 + 144*x + 2016*x^2 + 5376*x^3 + 2304x^4)/(1 - 4*x)^(9/2) = 1 + 162*x + 4806*x^2 + 71892*x^3 + ....
Note that the zeros of the polynomial 1 + 144*x^2 + 2016*x^4 + 5376*x^6 + 2304*x^8 = 1/2*((1 + 2*x)^9 + (1 - 2*x)^9), are given by 1/2*cot(k*Pi/9)*i for 1 <= k <= 8. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(9/2) + ((1 - 2*x)/(1 + 2*x))^(9/2) ) = 1 + 162*x^2 + 4806*x^4 + 71892*x^6 + ....

A285045 Expansion of cosh(7*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 98, 1862, 19796, 160454, 1114428, 7008540, 41132520, 229435206, 1230873644, 6403088692, 32488200472, 161473267228, 788758622680, 3796375603320, 18040943163600, 84786596572230, 394599588033420, 1820669979129540, 8335975464699960
Offset: 0

Views

Author

Peter Bala, Apr 10 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285044, A285046.

Programs

  • Maple
    seq(1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n), n = 0..20);
  • Mathematica
    CoefficientList[Series[Cosh[7*ArcTanh[2Sqrt[x]]],{x,0,20}],x] (* Harvey P. Dale, Jun 07 2024 *)

Formula

a(n) = 1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n).
O.g.f. cosh(7*arctanh(2*sqrt(x))) = (1 + 84*x + 560*x^2 + 448*x^3)/(1 - 4*x)^(7/2) = 1 + 98*x + 1862*x^2 + 19796*x^3 + ....
Note that the zeros of the polynomial 1 + 84*x^2 + 560*x^4 + 448*x^6 = 1/2*((1 + 2*x)^7 + (1 - 2*x)^7), are given by 1/2*cot(k*Pi/7)*i for 1 <= k <= 6. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(7/2) + ((1 - 2*x)/(1 + 2*x))^(7/2) ) = 1 + 98*x^2 + 1862*x^4 + 19796*x^6 + ....
D-finite with recurrence: n*(2*n-1)*a(n) +2*(-8*n^2+16*n-57)*a(n-1) +16*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

A383928 Expansion of g.f. cosh(9*arctanh(4*sqrt(x))).

Original entry on oeis.org

1, 648, 76896, 4601088, 194102784, 6662320128, 199818854400, 5451206492160, 138644854013952, 3341194489757696, 77151510667984896, 1720777996555517952, 37293854107184922624, 788969931176505507840, 16350749459194860011520, 332885987884833366343680, 6673058165121160335851520
Offset: 0

Views

Author

Karol A. Penson, May 15 2025

Keywords

Crossrefs

Cf. A285043, A285044, A285045, A285046.

Programs

  • Maple
    seq(coeff(series((589824*x^4 + 344064*x^3 + 32256*x^2 + 576*x + 1)/(-16*x + 1)^(9/2), x, 17), x, k), k=0..16);

Formula

a(n) = 4^n*(105 + 400*n + 3392*n^2 + 512*n^3 + 4096*n^4)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 576*x + 32256*x^2 + 344064*x^3 + 589824*x^4)/(-16*x + 1)^(9/2).
E.g.f.: 134217728*((x^4 + 41/128*x^3 + 425/16384*x^2 + 525/1048576*x + 105/134217728)*BesselI(0, 8*x) + x*BesselI(1, 8*x)*(x^3 + 33/128*x^2 + 193/16384*x + 25/1048576))*exp(8*x)/105.

A384417 Expansion of g.f. cosh(9*arctanh(8*sqrt(x))).

Original entry on oeis.org

1, 2592, 1230336, 294469632, 49690312704, 6822215811072, 818458027622400, 89312567167549440, 9086229152658358272, 875874088323041460224, 80899222450192930308096, 7217466034064795168145408, 625687045828728598806134784, 52946875811413468120885493760, 4389120887020725640048536453120
Offset: 0

Views

Author

Karol A. Penson, May 28 2025

Keywords

Crossrefs

Cf. A383928, A384335, A285043, A285044, A285045, A285046.

Programs

  • Maple
    seq(coeff(series((1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2), x, 15), x, k), k=0..14);
  • Mathematica
    CoefficientList[Series[Cosh[9*ArcTanh[8*Sqrt[x]]],{x,0,14}],x] (* Stefano Spezia, May 29 2025 *)

Formula

a(n) = 16^n*(105 + 400*n + 3392*n^2 + 512*n^3 + 4096*n^4)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2).
E.g.f.: exp(32*x)*((105 + 512*x*(269 + 256*x*(73 + 512*x)))*BesselI(0, 32*x) + 512*x*(25 + 256*x*(65 + 512*x))*BesselI(1, 32*x))/105 + (131072*x*hypergeom([3/2, 2, 2], [1, 1, 1], 64*x))/105.

A384542 Expansion of g.f. sinh(7*arctanh(14*sqrt(x)))/(98*sqrt(x)).

Original entry on oeis.org

1, 1666, 1090054, 485318932, 176760328262, 56963958713340, 16909346921973660, 4732136004374122344, 1266899066122354262598, 327667319343098397330668, 82435716917761454374571444, 20275150472587631020453400984, 4893425028040341625551135687452, 1162305136998381407493307772297560
Offset: 0

Views

Author

Karol A. Penson, Jun 02 2025

Keywords

Crossrefs

Cf. A383928, A384335, A285043, A285044, A285045, A285046, A384417.

Programs

  • Mathematica
    a[n_]:=49^n*(105 + 464*n + 704*n^2 + 512*n^3)*(2*n)!/(105*(n!)^2); Array[a,14,0] (* Stefano Spezia, Jun 02 2025 *)

Formula

a(n) = 49^n*(105 + 464*n + 704*n^2 + 512*n^3)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 980*x + 115248*x^2 + 1075648*x^3)/(-196*x + 1)^(7/2).
E.g.f.: exp(98*x)*(BesselI(0, 98*x)*(275365888*x^3 + 5444096*x^2 + 23520*x + 15) + 224*x*BesselI(1, 98*x)*(1229312*x^2 + 18032*x + 29))/15.
Showing 1-6 of 6 results.

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