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.

A109747 E.g.f.: exp(-exp(-x)+1+x).

Original entry on oeis.org

1, 2, 3, 3, 2, 3, 5, -4, 5, 55, -212, 201, 2381, -15350, 35183, 145359, -1821438, 8117231, -521487, -278996548, 2261959961, -7554900397, -34727188796, 690775844605, -4901767330647, 10921820177234, 179314430713387, -2668801066419061, 18150518618843778
Offset: 0

Views

Author

Franklin T. Adams-Watters and Vladeta Jovovic, Aug 10 2005

Keywords

Comments

Equals double binomial transform of A014182. - Gary W. Adamson, Dec 31 2008

Examples

			G.f. = 1 + 2*x + 3*x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 5*x^6 - 4*x^7 + 5*x^8 + 55*x^9 + ...

Crossrefs

Cf. A080094.
Cf. A014182. - Gary W. Adamson, Dec 31 2008

Programs

  • Maple
    G:=exp(-exp(-x)+1+x): Gser:=series(G,x=0,32): seq(n!*coeff(Gser,x,n),n=0..28); # Emeric Deutsch, Apr 10 2006
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[-Exp[-x]+1+x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 22 2018 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A000522(k).
G.f. = (1 - x^2 * Sum_{k>0} k * x^k / ((1 + x) * (1 + 2*x) + ... (1 + k*x))) / (1 - x)^2. - Michael Somos, Nov 07 2014
G.f.: 1/(1-x*Q(0)), where Q(k)= 1 + x/(1 - x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
G.f.: 1/W(0), where W(k) = 1 - x - x/(1 + x*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2014
a(n) = exp(1) * (-1)^n * Sum_{k>=0} (-1)^k * (k - 1)^n / k!. - Ilya Gutkovskiy, Dec 20 2019

Extensions

More terms from Emeric Deutsch, Apr 10 2006

A080093 Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=x(n).

Original entry on oeis.org

0, 1, 1, 2, 11, 41, 81, 715, 3425, 8861, 98253, 580317, 1816640, 24011157, 166888165, 608035190, 9264071767, 73600798037, 304238004061, 5224266196935, 46499892038437, 214184962059157, 4078345814329009, 40073660040755337
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 28 2003

Keywords

Examples

			Values of sum(k>=0,k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n) are given by n=1: (1/e)/2 = 0.183939720585721160..., n=2: (e - 3/e)/8 = 0.201830438118089783..., n=3: (e + 3/e)/16 = 0.238870009498335762..., n=4: (2e - 1/e)/16 = 0.316792763484165509..., n=5: (11e + 3/e)/64 = 0.484449038071309758..., n=6: (41e - 5/e)/128 = 0.856329357507528461..., n=7: (81e - 2/e)/128 = 1.71441460330343577..., n=8: (715e - 5/e)/512 = 3.79244552762179713..., n=9: (3425e + 55/e)/1024 = 9.11166858568033130..., n=10: (8861e + 106/e)/1024 = 23.5602446315818092...

Crossrefs

Cf. A080094, A080095, A079750 - A079756.

A080095 Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=z(n).

Original entry on oeis.org

2, 8, 16, 16, 64, 128, 128, 512, 1024, 1024, 4096, 8192, 8192, 32768, 65536, 65536, 262144, 524288, 524288, 2097152, 4194304, 4194304, 16777216, 33554432, 33554432, 134217728, 268435456, 268435456, 1073741824, 2147483648
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 28 2003

Keywords

Links

Crossrefs

Cf. A080093, A080094.

Programs

Formula

a(n) = 2^b(n) and {b(n)}={1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, ..} where b(3n-2)=3n-2, b(3n-1)=3n, b(3n)=b(3n+1)=3n+1, for n>0.
Showing 1-3 of 3 results.

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