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.

A073478 Expansion of (1+x)^(1/(1-x)).

Original entry on oeis.org

1, 1, 2, 9, 44, 290, 2154, 19026, 186752, 2070792, 25119720, 334960560, 4824346152, 75100568088, 1250180063664, 22235660291880, 419595248663040, 8388866239417920, 176823515257447104, 3923498370610292544
Offset: 0

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Examples

			E.g.f.: (1+x)^(1/(1-x)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 44*x^4/4! + 290*x^5/5! + 2154*x^6/6! + 19026*x^7/7! + 186752*x^8/8! + 2070792*x^9/9! + ...
which may be written as
(1+x)^(1/(1-x)) = exp(x + x^2*(1+x)/2 + x^3*(1+x+x^2)/3 + x^4*(1+x+x^2+x^3)/4 + x^5*(1+x+x^2+x^3+x^4)/5 + ... + x^n*((1-x^n)/(1-x))/n + ...).

Links

Crossrefs

Cf. A007113, A007120, A087761.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)^(1/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)
  • PARI
    {a(n)=n!*polcoeff((1+x +x*O(x^n))^(1/(1-x)),n)} \\ Paul D. Hanna, Jan 08 2014
  • PARI
    {a(n)=local(A);A=exp(sum(m=1,n,sum(k=1,m,-(-1)^k/k)*x^m)+x*O(x^n)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Jan 08 2014

Formula

E.g.f.: exp( Sum_{n>=1} x^n * Sum_{k=1..n} -(-1)^k/k ). - Paul D. Hanna, Jan 08 2014
E.g.f.: exp( Sum_{n>=1} x^n * ((1-x^n)/(1-x)) / n ). - Paul D. Hanna, Nov 24 2024
a(n) ~ (log(2))^(1/4) * exp(2*sqrt(n*log(2)) - n - 1/2) * n^(n-1/4). - Vaclav Kotesovec, Apr 21 2014

Extensions

More terms from Robert G. Wilson v, Aug 28 2002

A073479 Expansion of e.g.f.: (1-x)^(-1-x).

Original entry on oeis.org

1, 1, 4, 15, 80, 490, 3534, 28938, 266048, 2710440, 30311640, 369127440, 4862219592, 68881435896, 1044331262688, 16872336545400, 289380447338880, 5251237965683520, 100519388543098944, 2024241909160239936, 42780009017657888640, 946724781741392908800
Offset: 0

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 80*x^4/4! + 490*x^5/5! +...
Explicit expressions for the e.g.f.:
(1-x)^(-1-x) = 1 + (1+x)*x + (1+x)(2+x)*x^2/2! + (1+x)(2+x)(3+x)*x^3/3! +... - _Paul D. Hanna_, Nov 01 2010
(1-x)^(-1-x) = exp(x + 3*x^2/2 + 5*x^3/6 + 7*x^4/12 + 9*x^5/20 + 11*x^6/30 +...). - _Paul D. Hanna_, Sep 27 2014

Links

Crossrefs

Cf. A007113, A007120.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 30 2018
  • Maple
    S:= series((1-x)^(-1-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Apr 20 2017
  • Mathematica
    CoefficientList[ Series[(1 - x)^(-1 - x), {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,prod(k=1,m,k+x)*x^m/m!)+x*O(x^n),n)} \\ Paul D. Hanna, Nov 01 2010
  • PARI
    {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-x),n)} \\ Paul D. Hanna, Nov 01 2010

Formula

(1-x)^(-1-x) = Sum_{n>=0} (Product_{k=1..n} (k+x)) * x^n/n!. [Paul D. Hanna, Nov 01 2010]
E.g.f.: (1-x)^(-1-x) = 1+(x*(1+x))/(Q(0)-x*(1+x)); Q(k)=(1+x)*k+1+x+(x^2)-x*(k+1)*(k+2+x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2011
E.g.f.: 1 + x*(Q(0) - 1)/(x-1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ n! * (n - log(n) + 1 - gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 29 2013
E.g.f.: exp( x + Sum_{n>=2} (2*n-1) * x^n / (n*(n-1)) ). - Paul D. Hanna, Sep 27 2014

A247212 Exponential generating function = (1+x)^(1+x^2).

Original entry on oeis.org

1, 1, 0, 6, 12, -20, 420, -252, -336, 66960, -368640, 2328480, 2898720, -117767520, 1720764864, -12297479040, 58230547200, 312819736320, -9239378296320, 128087696977920, -1154590730496000, 7050771080478720, 398679450301440, -591762353886950400, 6580219687752775680
Offset: 0

Views

Author

Franklin T. Adams-Watters, Nov 29 2014

Keywords

Examples

			(1+x)^(1+x^2) = 1+x+x^3+(1/2)*x^4-(1/6)*x^5+(7/12)*x^6-(1/20)*x^7-(1/120)*x^8+(31/168)*x^9-(32/315)*x^10+(7/120)*x^11+...

Crossrefs

Cf. A005727, A007120, A247256.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1+x)^(1+x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 24 2019 *)
  • PARI
    Vec(serlaplace((1+x+O(x^25))^(1+x^2)))

A247256 Exponential generating function = (1+x)^(1-x^2).

Original entry on oeis.org

1, 1, 0, -6, -12, 20, 300, 252, -3024, -36720, 106560, 110880, 12212640, -125629920, 1005286464, -16865735040, 252900345600, -3575747185920, 58447092395520, -1014901444454400, 18218754479923200, -346655486035998720, 6952232946445839360, -145913061049673702400, 3205301440394904238080
Offset: 0

Views

Author

Franklin T. Adams-Watters, Nov 29 2014

Keywords

Examples

			(1+x)^(1-x^2) = 1 + x - x^3 - 1/2*x^4 + 1/6*x^5 + 5/12*x^6 + 1/20*x^7 - 3/40*x^8 - 17/168*x^9 + 37/1260*x^10 + 1/360*x^11 + 257/10080*x^12 - ...

Crossrefs

Cf. A005727, A007120, A247212.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)^(1-x^2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 30 2014 *)
  • PARI
    Vec(serlaplace((1+x+O(x^25))^(1-x^2)))

Formula

a(n) ~ 2 * (-1)^n * n! / n^2 * (1 + 4*log(n)/n). - Vaclav Kotesovec, Nov 30 2014

A347978 E.g.f.: 1/(1 + x)^(1/(1 - x)).

Original entry on oeis.org

1, -1, 0, -3, 4, -30, 186, -630, 11600, -26712, 1005480, -2581920, 117196872, -485308824, 17734457664, -131070696120, 3387342915840, -43890398953920, 801577841697216, -17363169328243392, 233460174245351040, -7968629225100337920, 84363134551361043840
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2021

Keywords

Crossrefs

Cf. A005727, A007120, A008405, A024167, A058312, A058313, A073478, A087761.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A024167[n_] := n! Sum[(-1)^(k + 1)/k, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] A024167[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ Michel Marcus, Sep 22 2021

Formula

E.g.f.: exp( Sum_{k>=1} x^k * Sum_{j=1..k} (-1)^j / j ).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A024167(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * A073478(k) * a(n-k).
Showing 1-5 of 5 results.

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