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.

A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

Original entry on oeis.org

1, 1, 5, 47, 660, 12414, 293552, 8374806, 280064600, 10747277832, 465597887592, 22479948822792, 1197060450322800, 69699159437088960, 4405397142701855232, 300408348609092268144, 21983809533066553697280
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 47*x^3/3! + 660*x^4/4! +... [_Paul D. Hanna_, Aug 28 2008]

Links

Crossrefs

Cf. A052819. [From Paul D. Hanna, Aug 28 2008]
Cf. A052894, A198860.

Programs

  • Maple
    spec := [S,{C=Cycle(B),S=Sequence(C),B=Prod(S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x + x*Log[1-x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Jan 08 2014 *)
  • Maxima
    a(n):=sum(binomial(n+k,n)*k!*sum((-1)^(n+j)/(k-j)!*sum(stirling1(n,j-i)/i!,i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, May 09 2013 */
  • PARI
    a(n)=n!*polcoeff((1/x)*serreverse(x+x*log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Aug 28 2008

Formula

From Paul D. Hanna, Aug 28 2008: (Start)
E.g.f. satisfies: A(x*(1 + log(1-x))) = 1/(1 + log(1-x)).
E.g.f. satisfies: A(x) = 1/(1 + log(1 - x*A(x))).
E.g.f.: A(x) = (1/x)*Series_Reversion[x + x*log(1-x)]. (End)
a(n)=sum(k=0..n, binomial(n+k,n)*k!*sum(j=0..k, (-1)^(n+j)/(k-j)!*sum(i=0..j, stirling1(n,j-i)/i!)))/(n+1); [Vladimir Kruchinin, May 09 2013]
a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(exp(2)) = 1.5571455989976114... - Vaclav Kotesovec, Jan 08 2014
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * |Stirling1(n,k)|. - Seiichi Manyama, Nov 06 2023

Extensions

New name using e.g.f., Vaclav Kotesovec, Jan 08 2014

A367136 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^2)).

Original entry on oeis.org

1, 1, 5, 50, 764, 15804, 413426, 13094864, 487323000, 20844584760, 1007739144312, 54343954158240, 3234285062655984, 210581685526690464, 14889759832273000320, 1136236597054802033664, 93074880409847175490560, 8146156595011083708521472
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A006252, A198860, A367137.
Cf. A367134, A367138.

Programs

  • Mathematica
    Table[1/(2*n+1)! * Sum[(2*n+k)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k)!*stirling(n, k, 1))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * Stirling1(n,k).
a(n) ~ LambertW(2*exp(1))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(1)))) * exp(n) * (2 - LambertW(2*exp(1)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367137 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^3)).

Original entry on oeis.org

1, 1, 7, 101, 2248, 68024, 2608940, 121316796, 6633841608, 417181294704, 29665022908992, 2353675598751960, 206145540193974288, 19755830347828845360, 2056381966404400741920, 231034314706671715165824, 27865886237401381188422400, 3591366670194210901813749120
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A006252, A198860, A367136.
Cf. A367135, A367139.

Programs

  • Mathematica
    Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k)!*stirling(n, k, 1))/(3*n+1)!;

Formula

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling1(n,k).
a(n) ~ LambertW(3*exp(2))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(2)))) * exp(n) * (3 - LambertW(3*exp(2)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A370937 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - log(1+3*x)/3) ).

Original entry on oeis.org

1, 1, 1, 3, 12, 54, 432, 2862, 29880, 311688, 3530952, 52947432, 694960560, 12339656640, 208855024128, 3885592056624, 84031138091520, 1688108258868480, 41851910546369280, 986544325475565696, 25610732492679696384, 720669291974958124800, 19681263432530494848000
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A198860, A370936.

Programs

  • Mathematica
    a[n_]:=Sum[3^(n-k)*(n+k)!*StirlingS1[n, k],{k,0,n}]/(n+1)!; Array[a,23,0] (* Stefano Spezia, Apr 20 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-log(1+3*x)/3))/x))
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*(n+k)!*stirling(n, k, 1))/(n+1)!;

Formula

a(n) = (1/(n+1)!) * Sum_{k=0..n} 3^(n-k) * (n+k)! * Stirling1(n,k).

A370936 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - log(1+2*x)/2) ).

Original entry on oeis.org

1, 1, 2, 8, 48, 384, 3872, 47088, 671360, 10985088, 202927872, 4178030592, 94874787840, 2355758714880, 63498696376320, 1846607063998464, 57630620308930560, 1921296165774950400, 68145277700464312320, 2562234152415762972672, 101801592691389968154624
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A198860, A370937.

Programs

  • Mathematica
    a[n_]:=Sum[2^(n-k)*(n+k)!*StirlingS1[n, k],{k,0,n}]/(n+1)!; Array[a,21,0] (* Stefano Spezia, Apr 20 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-log(1+2*x)/2))/x))
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*(n+k)!*stirling(n, k, 1))/(n+1)!;

Formula

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * Stirling1(n,k).
a(n) ~ 2^(2*n + 1) * LambertW(exp(-1))^n * n^(n-1) / (sqrt(1 + LambertW(exp(-1))) * exp(n) * (1 - LambertW(exp(-1)))^(2*n + 1)). - Vaclav Kotesovec, Mar 06 2024
Showing 1-5 of 5 results.

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