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.

A141308 Euler transform of A141307.

Original entry on oeis.org

2, 7, 36, 283, 2898, 36169, 524976, 8659186, 159736316, 3257811334, 72797444280, 1769125982092, 46466434382032, 1311960028913384, 39633438764146568, 1275742281105759813, 43593785716301112538, 1576217593145774955007
Offset: 1

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Author

Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Keywords

Comments

Dimensions of the graded components of the domain of cocommutativity of the Hopf algebra of free quasi-symmetric functions of level 2.

Links

Crossrefs

Cf. A003319, A141307.

Programs

  • Maple
    EULER([seq(c(n),n=1..20)]); # where c(n) is A141307.
  • Mathematica
    Clear[c]; c[0]=0; c[n_] := c[n] = n! - Sum[k!*c[n-k], {k, 1, n-1}]; Rest[CoefficientList[Series[Product[1/(1 - x^k)^(2^k * c[k]), {k, 1, 20}], {x, 0, 20}], x]] (* Vaclav Kotesovec, Aug 07 2015 *)

Formula

a(n) ~ n! * 2^n * (1 - 1/n - 5/(4*n^2) - 21/(4*n^3) - 469/(16*n^4) - 3375/(16*n^5) - 118775/(64*n^6) - 1227535/(64*n^7) - 29026957/(128*n^8) - 385505947/(128*n^9) - 22698285665/(512*n^10)). - Vaclav Kotesovec, Aug 07 2015

A141309 INVERTi transform of A141308.

Original entry on oeis.org

2, 3, 16, 158, 1796, 24250, 372656, 6429927, 122956714, 2581840735, 59084565968, 1464445686726, 39101805324620, 1119446453417490, 34220008384633264, 1112764183812405300, 38362648544330997488, 1397884269388233025156
Offset: 1

Views

Author

Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Keywords

Comments

Number of generators of degree n of the (free) primitive Lie algebra of the Hopf algebra of free quasi-symmetric functions of level 2.

Links

Crossrefs

Cf. A003319, A141307, A141308.

Programs

  • Maple
    INVERTi([seq(d(n,n=1..20)]); # where d(n) is A141308
  • Mathematica
    terms = 18;
c[0] = 0; c[n_] := c[n] = n! - Sum[k! c[n-k], {k, 1, n-1}];
s = (Product[1/(1-x^k)^(2^k c[k]), {k, 1, terms+1}] + O[x]^(terms+1)-1)/x;
CoefficientList[-1/(1 + x s) + O[x]^(terms+1), x] // Rest (* Jean-François Alcover, Feb 13 2019 *)

A305535 Expansion of 1/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

Comments

Invert transform of A000165, shifted right one place.

Links

Crossrefs

Cf. A000165, A051295, A051296, A112934, A141307, A292778.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(k - 1) (k - 1)! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) ~ 2^(n-1) * (n-1)!. - Vaclav Kotesovec, Sep 18 2021
Showing 1-3 of 3 results.

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

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