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.

A052873 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x*A(x))).

Original entry on oeis.org

1, 1, 5, 46, 629, 11496, 263857, 7301680, 236748969, 8806142080, 369714769181, 17296339048704, 892335712777885, 50333180563864576, 3081739132775658825, 203555129140352505856, 14428195498061848405073, 1092403962489972428144640, 87990832863810814525250869
Offset: 0

Views

Author

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

Keywords

Comments

Previous name was: A simple grammar.

Links

Crossrefs

Cf. A052868, A088690, A161630.
Cf. A382036, A382037, A382038.

Programs

  • Maple
    spec := [S,{C=Sequence(B,1 <= card),S=Set(C),B=Prod(Z,S)},labeled]:
seq(combstruct[count](spec,size=n), n=0..20);
# Alternatively:
a := n -> `if`(n=0,1, n!*hypergeom([1-n],[2],-n-1)):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Apr 20 2016
  • Mathematica
    Table[Sum[(n+1)^(k-1)*n!/k!*Binomial[n-1,k-1],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 08 2014 *)
  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,(n+1)^(k-1)*n!/k!*binomial(n-1,k-1)))} \\ Paul D. Hanna, Sep 08 2012
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^(m-1)*x^m/m!/(1-x*A+x*O(x^n))^m));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 08 2012

Formula

E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z)).
1 = Sum_{n>=0} a(n)*exp((n+1)*x/(x-1))*x^n/n!. - Vladeta Jovovic, Jul 20 2005
a(n) = Sum_{k=0..n} (n+1)^(k-1)*n!/k!*binomial(n-1,k-1). - Vladeta Jovovic, Jul 02 2006
E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! / (1-x*A(x))^n. - Paul D. Hanna, Sep 08 2012
Equivalently:
E.g.f. satisfies: A(x) = exp( x*A(x)/(1 - x*A(x)) ). - Olivier Gérard, Dec 29 2013
a(n) ~ (sqrt(5)-1) * 2^(n-1/2) * n^(n-1) * exp((sqrt(5)-1 + (sqrt(5)-3)*n)/2) / (5^(1/4) * (3-sqrt(5))^(n+1/2)). - Vaclav Kotesovec, Jan 08 2014
a(n) = n!*hypergeom([1-n],[2],-n-1) for n >= 1. - Peter Luschny, Apr 20 2016
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x) ) ). - Seiichi Manyama, Mar 15 2025

Extensions

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

A382032 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x))^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 55, 937, 21741, 639841, 22839139, 958882289, 46304377849, 2528571710881, 154076164781991, 10364272238514217, 762867688235619877, 60989719558159065857, 5263030218009265964011, 487578723768665716788961, 48266847740986728218648433, 5084697384633390178057209793
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161630, A382033, A382034.
Cf. A000108, A377553, A382036.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(2*n, k)/(n-k-1)!));

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377553. F(x) = log(A(x))/x = C(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^2 ) ).

A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 160, 4325, 157896, 7280077, 406085632, 26599741065, 2001864880000, 170236619802161, 16144762562002944, 1689534516295056301, 193403842876754728960, 24040636567791329323125, 3224829927677539092791296, 464325325579881390473331473, 71428455280041816247241637888
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A052873, A382036, A382038.
Cf. A001764, A377830, A382033.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(3*n, k)/(n-k-1)!));

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^3).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^3 ) ).

A382038 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^4) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 11, 244, 8285, 381096, 22175167, 1562582848, 129381990201, 12313784396800, 1324663415429651, 158957183013686784, 21051725357219126869, 3050121640032545419264, 479928476696367747954375, 81499293517054315684642816, 14856515462975583258374526833, 2893604521320117995839047401472
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A052873, A382036, A382037.
Cf. A002293, A382034.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(4*n, k)/(n-k-1)!));

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^4).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^4 ) ).

A382086 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 52, 845, 18816, 533617, 18404800, 748039833, 35016198400, 1855389108221, 109781344134144, 7174844881882405, 513331696318615552, 39905830821183755625, 3349445733955326754816, 301886246619209909215793, 29080090017105458412257280, 2981488457660004727761477493
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382087, A382088, A382089.
Cf. A000108, A377831, A382036.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(n+k-1, k)/(n-k-1)!));

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x) * exp(-x) ) ).
a(n) ~ phi^(3*n - 3/2) * n^(n-1) / (5^(1/4) * exp((n - 1/phi)/phi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 15 2025
Showing 1-5 of 5 results.

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