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 1-3 of 3 results.

A002017 Expansion of e.g.f. exp(sin(x)).

Original entry on oeis.org

1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, 5973, 309376, 1237173, -2917888, -52635599, -163782656, 1126610929, 12716052480, 20058390573, -495644917760, -3920482183827, 4004259037184, 256734635981833, 1359174582304768
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of set partitions of 1..n into odd parts with an even number of parts of size == 3 (mod 4), minus the number of such partitions with an odd number of parts of size == 3 (mod 4). - Franklin T. Adams-Watters, Apr 29 2010

Examples

			For n=6, there are 6 partitions with part sizes [5,1], 10 with sizes [3^2], 20 with sizes [3,1^3], and 1 with sizes [1^6]; 6 + 10 - 20 + 1 = -3. - _Franklin T. Adams-Watters_, Apr 29 2010

References

  • CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|.
Cf. A003724. - Franklin T. Adams-Watters, Apr 29 2010
Cf. A047687, A047688 for numerators & denominators of the series of exp(sin(x)) at x = 0.

Programs

  • Mathematica
    max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* Jean-François Alcover, Jun 26 2013 *)
  • Maxima
    a(n):=2*sum((sum((2*i-n+2*j)^n*binomial(n-2*j,i)*(-1)^(n-j-i),i,0,(n-2*j)/2))/(2^(n-2*j)*(n-2*j)!),j,0,(n-1)/2); /* Vladimir Kruchinin, Jun 10 2011 */
  • Maxima
    a(n):=if n=0 then 1 else (n-1)!*sum((-1)^(k)/(2*k)!*a(n-2*k-1)/(n-2*k-1)!,k,0,(n-1)/2); /* Vladimir Kruchinin, Feb 25 2015 */
  • PARI
    my(x='x+O('x^33)); Vec(serlaplace(exp(sin(x)))) \\ Joerg Arndt, Apr 01 2017

Formula

a(n) = 2*Sum_{j=0..(n-1)/2} Sum_{i=0..(n-2*j)/2} (2*i-n+2*j)^n*C(n-2*j,i)*(-1)^(n-j-i)/(2^(n-2*j)*(n-2*j)!), n>0, a(0)=1. - Vladimir Kruchinin, Jun 10 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A003724. - Peter Bala, Dec 06 2011
E.g.f.: 1 + sin(x)/T(0), where T(k) = 4*k+1 - sin(x)/(2 + sin(x)/(4*k+3 - sin(x)/(2 + sin(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
E.g.f.: 2/Q(0), where Q(k) = 1 + 1/( 1 - sin(x)/( sin(x) - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
E.g.f.: E(0)-1, where E(k) = 2 + sin(x)/(2*k + 1 - sin(x)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = (n-1)!*Sum_{k=0..(n-1)/2} ((-1)^k/(2*k)!)*a(n-2*k-1)/(n-2*k-1)!, a(0)=1. - Vladimir Kruchinin, Feb 25 2015

Extensions

Extended with signs by Christian G. Bower, Nov 15 1998

A296466 Expansion of e.g.f. arcsinh(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 0, 8, 56, 8000, 342144, 68623488, 8295676416, 2411783847936, 584142614728704, 240810283258527744, 96772676958798741504, 54867909992513301282816, 32661008325245409302937600, 24691868812821871169667072000, 20243305132513358736699378892800, 19829947398943934886214249532620800
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsinh(arcsin(x)) = x/1! + 8*x^5/5! + 56*x^7/7! + 8000*x^9/9! + 342144*x^11/11! + 68623488*x^13/13! + ...

Crossrefs

Cf. A001818, A003722, A012248, A296464.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[Log[Sqrt[1 - Log[I x + Sqrt[1 - x^2]]^2] - I Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: log(sqrt(1 - log(i*x + sqrt(1 - x^2))^2) - i*log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).
a(n) ~ 2 * (2*n)! / sqrt(Pi*(4 + Pi^2)*n). - Vaclav Kotesovec, Dec 13 2017

A296791 Expansion of e.g.f. sech(x*sec(x)) (even powers only).

Original entry on oeis.org

1, -1, -7, -1, 3121, 132959, -1261591, -889217057, -79029091743, 5889540654911, 3289057601679065, 395957721046153023, -120519140613246313327, -71865162873642033099361, -9267049529998625177827639, 8376363338336819515365004319, 5693280488360087435524724806849
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			sech(x*sec(x)) = 1 - x^2/2! - 7*x^4/4! - x^6/6! + 3121*x^8/8! + ...

Crossrefs

Cf. A000364, A003721, A003722, A009008, A009009, A009010, A009011, A009015, A009016, A009118, A009562, A009765, A102072, A102075, A296731, A296740, A296790.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Sech[x Sec[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)] sech(x*sec(x)).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.