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 1-5 of 5 results.

A003727 Expansion of e.g.f. exp(x * cosh(x)).

Original entry on oeis.org

1, 1, 1, 4, 13, 36, 181, 848, 3865, 23824, 140521, 871872, 6324517, 44942912, 344747677, 2860930816, 23853473329, 213856723200, 1996865965009, 19099352929280, 193406280000061, 2010469524579328, 21615227339380357, 242177953175506944
Offset: 0

Views

Author

R. H. Hardin

Keywords

References

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

Links

Crossrefs

Cf. A009233, A191509.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Cosh(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 09 2018
  • Mathematica
    CoefficientList[Series[E^(x*Cosh[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 05 2014 *)
Table[Sum[BellY[n, k, Mod[Range[n], 2] Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
  • Maxima
    a(n):=sum(if n=k then n! else 1/2^k*sum(binomial(n,k)*binomial(k,i)*(k-2*i)^(n-k),i,0,k),k,1,n); /* Vladimir Kruchinin, Aug 22 2010 */
  • PARI
    x='x+O('x^66);
Vec(serlaplace(exp( x * cosh(x) )))
/* Joerg Arndt, Sep 14 2012 */

Formula

a(n) = Sum_{k=1..n} (if n=k then n! otherwise (1/2)^k*Sum_{i=0..k} binomial(n,k)* binomial(k,i)*(k-2*i)^(n-k)), n>0. - Vladimir Kruchinin, Aug 22 2010
a(n) ~ exp(r*cosh(r)-n) * n^n / (r^n * sqrt(3+(r*(r^2-2)*cosh(r))/n)), where r is the root of the equation r*(cosh(r)+r*sinh(r)) = n. - Vaclav Kotesovec, Aug 05 2014
a(n)^(1/n) ~ n*exp(1/(2*LambertW(sqrt(n/2)))-1) / (2*LambertW(sqrt(n/2))). - Vaclav Kotesovec, Aug 05 2014
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extensions

Extended and formatted by Olivier Gérard, Mar 15 1997

A009214 E.g.f. exp(x*sin(x)) (even powers only).

Original entry on oeis.org

1, 2, 8, 6, -792, -10790, 281940, 13531350, -260660176, -33714262350, 550333492140, 158933551076014, -2777269276818168, -1301993178430302774, 33725324008920743108, 17091479764089813623430
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A009233.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[Exp[x Sin[x]], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
  • Maxima
    a(n):=sum(binomial(2*n,k)*(sum((2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i),i,0,k/2))/(2^(k-1)),k,1,n); /* Vladimir Kruchinin, Jun 06 2011 */
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(n)=sum(k=1..n, binomial(2*n,k)*(i=0..k/2, sum((2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i)))/(2^(k-1))). - Vladimir Kruchinin, Jun 06 2011
a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k+1) * binomial(2*n-1,2*k-1) * k * a(n-k). - Ilya Gutkovskiy, Mar 10 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 28, 966, 62280, 6452650, 980531916, 205438870014, 56760128400016, 19994672935658322, 8746764024725937300, 4651991306703670964518, 2956156902003429777549144, 2212026607642404922284728826, 1925137044528752884360406444380, 1928103808741894922401976601295950
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A009233, A094088, A205571, A210672, A351762, A352250, A352252, A352253.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[1/(1 - x Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...

Crossrefs

Cf. A001818, A006228, A009214, A009233, A079484, A259647, A296787, A296789.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Exp[x ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(x + Sqrt[1 + x^2])^x, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arcsinh(x)).
a(n) ~ -(-1)^n * 2^(2*n) * n^(2*n-1) / exp(2*n + Pi/2). - Vaclav Kotesovec, Dec 21 2017

A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

Original entry on oeis.org

1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A000248, A000807, A003724, A003727, A005046, A009233, A351937, A352253.

Programs

  • Mathematica
    nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * k * a(n-k).
Showing 1-5 of 5 results.

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