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 1-7 of 7 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 + ....

A085840 Triangle read by rows: T(n,m) = 4^m * (2*n+1)! / ( (2*n - 2*m + 1)! * (2*m)! ), row n has n+1 terms.

Original entry on oeis.org

1, 1, 12, 1, 40, 80, 1, 84, 560, 448, 1, 144, 2016, 5376, 2304, 1, 220, 5280, 29568, 42240, 11264, 1, 312, 11440, 109824, 329472, 292864, 53248, 1, 420, 21840, 320320, 1647360, 3075072, 1863680, 245760
Offset: 0

Views

Author

Gary W. Adamson, Jul 05 2003

Keywords

Comments

Row n has the unsigned coefficients of a polynomial whose roots are 2*tan(Pi*k/(2n+1)) [for k = 1 to 2n].
Polynomial of row n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).

Examples

			1
x^2 - 12
x^4 - 40x^2 + 80
x^6 - 84x^4 + 560x^2 - 448
x^8 - 144x^6 + 2016x^4 - 5376x^2 + 2304
x^10 - 220x^8 + 5280x^6 - 29568x^4 + 42240x^2 - 11264
Polynomial #4 has eight roots: 2 tan (Pi*k/9) for k=1 to 8.

Crossrefs

Cf. A085841, A285043, A284044, A285045, A285046.

Programs

  • Maple
    for n from 0 to 10 do lprint(seq(4^k*binomial(2*n + 1, 2*k), k = 0..n)) end do; # Peter Bala, Apr 10 2017

Formula

From Peter Bala, Apr 10 2017: (Start)
O.g.f.: (1 - (1 - 4*x)*t)/(1 - 2*(1 + 4*x)*t + (1 - 4*x)^2*t^2) = 1 + (1 + 12*x)*t + (1 + 40*x + 80*x^2)*t^2 + ....
n_th row polynomial R(n,x) = 1/2*( (1 + 2*sqrt(x))^(2*n+1) + (1 - 2*sqrt(x))^(2*n+1) ). These polynomials occur in the expansion cosh((2*n + 1)*arctanh(2*x)) = R(n,x^2)/(1 - 4*x^2)^(n+1/2). See A285043 - A285046.
For n >= 1, R(n,x) = (1 - 4*x)^n( U(n,(1 + 4*x)/(1 - 4*x)) - U(n-1,(1 + 4*x)/(1 - 4*x)) ), where U(n,x) is the n-th Chebyshev polynomial of the second kind. (End)

Extensions

Edited by Don Reble, Nov 13 2005

A285044 Expansion of cosh(5*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 50, 550, 4020, 24710, 138012, 725340, 3655080, 17859270, 85230860, 399257716, 1842353240, 8396404380, 37868584600, 169278679800, 750923914320, 3308947546950, 14495583969900, 63172016823300, 274031830241400, 1183780040663220
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, A285045, A285046.

Programs

  • Maple
    seq(1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n), n = 0..20);

Formula

a(n) = 1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n).
O.g.f. cosh(5*arctanh(2*sqrt(x))) = (1 + 40*x + 80*x^2)/(1 - 4*x)^(5/2) = 1 + 50*x + 550*x^2 + 4020*x^3 + ....
Note that the zeros of the polynomial 1 + 40*x^2 + 80*x^4 = 1/2*((1 + 2*x)^5 + (1 - 2*x)^5), are given by 1/2*cot(k*Pi/5)*i for 1 <= k <= 4. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(5/2) + ((1 - 2*x)/(1 + 2*x))^(5/2) ) = 1 + 50*x^2 + 550*x^4 + 4020*x^6 + ....

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 1-7 of 7 results.

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