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.

A003703 Expansion of e.g.f. cos(log(1+x)).

Original entry on oeis.org

1, 0, -1, 3, -10, 40, -190, 1050, -6620, 46800, -365300, 3103100, -28269800, 271627200, -2691559000, 26495469000, -238131478000, 1394099824000, 15194495654000, -936096296850000, 29697351895900000, -819329864480400000, 21683886333440500000, -570263312237604700000
Offset: 0

Views

Author

R. H. Hardin, Simon Plouffe

Keywords

Examples

			1 - x^2 + 3*x^3 - 10*x^4 + 40*x^5 - 190*x^6 + 1050*x^7 - 6620*x^8 + ...

References

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

Links

Crossrefs

Cf. A009024, A009454.

Programs

  • Maple
    a:= n-> add(Stirling1(n, 2*k) * (-1)^(k), k=0..floor(n/2)):
seq(a(n), n=0..20);
  • Mathematica
    CoefficientList[Series[Cos[Log[1 + x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 16 2015 *)
Table[(-1)^n Im[Pochhammer[1-I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( cos( log( 1 + x + x * O(x^n))), n))} /* Michael Somos, Jul 26 2012 */
  • PARI
    a(n) = (-1)^n*(prod(k=0, n-1, I+k)+prod(k=0, n-1, -I+k))/2; \\ Seiichi Manyama, Oct 10 2022
  • Python
    from sympy.functions.combinatorial.numbers import stirling
def A003703(n): return sum(stirling(n,k<<1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

Formula

a(n) = Sum_{k=0..n-1} (-1)^(k+1)*T(n-k, k)*sin(Pi*(n-k-1)/2) + 0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry, Apr 18 2005
abs(a(n)) = abs(f(n)) with f(n)=Product_{k=1..n} i+k (where i^2=-1). - Yalcin Aktar, Jul 13 2009
a(n) = Sum_{k=0..floor(n/2)} Stirling1(n,2*k)*(-1)^k. - Vladimir Kruchinin, Jan 29 2011
a(n+2)= -a(n+1)*(2*n+1) - a(n)*(1+n^2), a(0)=1, a(1)=0. - Sergei N. Gladkovskii, Aug 17 2012
a(n) = (-1)^n * ( (i)n + (-i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit. - _Seiichi Manyama, Oct 10 2022
a(n) = Re(gamma(i+1)/gamma(i+1-n)). The imaginary part is A009454. - Colin Beveridge, Jul 30 2024

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A189815, A296980, A296981, A296982.

Programs

  • Maple
    a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ -(-1)^n * n^(n-1) / (exp(1) - 1)^(n - 1/2). - Vaclav Kotesovec, Mar 26 2019

A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A296435, A296979, A296981, A296982.

Programs

  • Maple
    a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

A296981 Expansion of e.g.f. arctan(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 0, 6, -22, -30, 952, -5656, -9952, 508320, -3874992, -20690208, 833780400, -7697940432, -52230156288, 2467649024640, -24686997151104, -329724479772288, 14493628861307136, -159114034671287040, -2682505451050592256, 126421889770129637376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arctan(log(1 + x)) = x^1/1! - x^2/2! + 6*x^4/4! - 22*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A003703, A003708, A009024, A009454, A009775, A010050, A104150, A110708, A296979, A296980, A296982.

Programs

  • Maple
    a:=series(arctan(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcTan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[(I/2) Log[1 - I Log[1 + x]] - (I/2) Log[1 + I Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ (-1)^(n+1) * (n-1)! * sin(n*(Pi-1)/2) / (2 - 2*cos(1))^(n/2). - Vaclav Kotesovec, Mar 26 2019

A296982 Expansion of e.g.f. arctanh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 4, -18, 118, -930, 8888, -98504, 1248784, -17790480, 281590032, -4901447232, 93064850448, -1914144990576, 42396742460928, -1006101059149440, 25466710774651776, -684902462140798848, 19503187752732408576, -586221766070655432960
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arctanh(log(1 + x)) = x^1/1! - x^2/2! + 4*x^3/3! - 18*x^4/4! + 118*x^5/5! - 930*x^6/6! + ...

Crossrefs

Cf. A001710, A003703, A003708, A009024, A009454, A009775, A010050, A104150, A202139, A296979, A296980, A296981.

Programs

  • Maple
    a:=series(arctanh(log(1+x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[ArcTanh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Log[1 + Log[1 + x]]/2 - Log[1 - Log[1 + x]]/2, {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-5 of 5 results.

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

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