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 1-10 of 24 results. Next

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

A377890 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 15, 211, 4433, 124741, 4412815, 188335981, 9421966209, 540884623753, 35054089163351, 2531882857204273, 201689970517618225, 17567711167993834381, 1661084543502646535967, 169448367505003640681221, 18550123929621138841581185, 2169272360350263071212545553
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377826, A377891.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*n-k+1)^(k-1) * binomial(2*n-k+1,n-k)/k!.
E.g.f.: (1/x) * Series_Reversion( x * (exp(-x) - x) ). - Seiichi Manyama, Dec 29 2024

A377829 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x)^2 ).

Original entry on oeis.org

1, 3, 25, 364, 7713, 216216, 7568041, 318256800, 15644919681, 880848974080, 55912403743161, 3951344780946432, 307737594185310625, 26190457718737019904, 2418475248758250599625, 240846113359411822759936, 25731326615411044591298049, 2935802801104074173428531200
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A088690, A377830.
Cf. A377827.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(x * A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n+2,n-k)/k!.
a(n) ~ (2*(1 + sqrt(2)))^(n + 1/2) * n^(n-1) / exp((2 - sqrt(2))*n + 1 - sqrt(2)). - Vaclav Kotesovec, Nov 09 2024

A377892 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * exp(x * A(x)^2).

Original entry on oeis.org

1, 2, 19, 352, 9885, 374486, 17907991, 1035748260, 70334590969, 5487022612810, 483655093883451, 47541690024105608, 5156503816883562325, 611769291578643110238, 78812382009451814165695, 10956572374811382997014796, 1634950184384280878142249969, 260653481562714033459279871250
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377893.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(2*n+1,n-k)/k!.

A377893 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * exp(x * A(x)^3).

Original entry on oeis.org

1, 2, 27, 742, 31337, 1796376, 130408603, 11472417104, 1186462228785, 141083381264896, 18966727953873371, 2844742575536036352, 470958524169176911513, 85307709328403287961600, 16782586179544965856158363, 3563492814539574559964993536, 812273035493592514001487147233
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377892.

Programs

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

Formula

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

A379456 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(x)) ).

Original entry on oeis.org

1, 2, 13, 151, 2573, 58221, 1648345, 56138461, 2236816825, 102135829609, 5259937376141, 301678137203433, 19072415186892325, 1317869007328182349, 98818139178323981473, 7991908824553634264101, 693473520767940388417265, 64266613784795934251538513
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A088690, A379846, A379847.
Cf. A108919, A161633.
Cf. A379699, A379700.

Programs

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

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (2*n-k+1)^k * binomial(n+1,n-k)/k!.
E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(2*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A088692 E.g.f: A(x) = f(x*A(x)), where f(x) = (1+2*x)*exp(x).

Original entry on oeis.org

1, 3, 23, 304, 5829, 147696, 4670371, 177383424, 7874174601, 400298556160, 22940919680271, 1463679309053952, 102911522568495757, 7906731860604186624, 659108356837269579675, 59252790438687592677376, 5714517052927568389576209, 588555892122678050845556736
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2003

Keywords

Comments

Radius of convergence of A(x): r = (1/4)*exp(-1/2) = 0.151632.., where A(r) = 2*exp(1/2) and r = lim_{n->infinity} (a(n)/a(n+1))*n.

Links

Crossrefs

Cf. A088690, A088693.

Programs

  • Maple
    f := n -> simplify(exp(-(1/4)*n-1/4)*2^n*factorial(n)*((5*n+1)*WhittakerM(-n, 1/2, (1/2)*n+1/2)-(2*n-2)*WhittakerM(1-n, 1/2, (1/2)*n+1/2))/(n+1)^2):
map(f, [$0..30]); # Robert Israel, Oct 08 2017
  • Mathematica
    Table[Sum[2^(n-k)*n^(k-2)*n!/k!*Binomial[n-1,k-1],{k,1,n}],{n,1,21}] (* Vaclav Kotesovec after Vladeta Jovovic, Jan 24 2014 *)
  • PARI
    a(n)=n!*polcoeff(((1+2*x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)

Formula

a(n) = n! * [x^n] ((1+2*x)*exp(x))^(n+1)/(n+1).
a(n) = Sum_{k=1..n} 2^(n-k)*n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - Vladeta Jovovic, Jun 19 2006
a(n) ~ 4^(n+1) * n^(n-1) / (sqrt(3) * exp(n/2-1/2)). - Vaclav Kotesovec, Jan 24 2014
a(n) = exp(-(n+1)/4)*2^n*n!*(n+1)^(-2)*((5*n+1)*WhittakerM(-n,1/2,(n+1)/2) - 2*(n-1)*WhittakerM(1-n,1/2,(n+1)/2)). - Robert Israel, Oct 08 2017

A377830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x)^3 ).

Original entry on oeis.org

1, 4, 45, 886, 25397, 963216, 45615553, 2595412240, 172624541769, 13150155923200, 1129371806449301, 107987110491257856, 11379014255782146685, 1310277285293012678656, 163703077517048727256425, 22057132253723442887059456, 3188342874266180285119069457, 492178313447920665621400780800
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A088690, A377829.

Programs

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

Formula

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

A377891 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 19, 364, 10665, 423056, 21221851, 1288931456, 91977076561, 7543664425216, 699290913249891, 72306463481715200, 8251192866018497401, 1030074741274860240896, 139650729116792108398891, 20432888021354725476499456, 3209204194084043665909835937, 538542735919965101952197525504
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377826, A377890.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (3*n-2*k+1)^(k-1)*binomial(3*n-2*k+1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n-2*k+1)^(k-1) * binomial(3*n-2*k+1,n-k)/k!.

A379884 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^2) - x).

Original entry on oeis.org

1, 2, 15, 223, 5045, 154161, 5949715, 277816813, 15234148585, 959821848433, 68333878996991, 5425649143910733, 475370226250388221, 45559752911807595865, 4741534923025152367627, 532526268840445510805341, 64198018232238090097818065, 8268729272698380485865553761
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377892, A379867.
Cf. A088690.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (n+k+1)^(k-1) * binomial(n+k+1,n-k)/k!.
Showing 1-10 of 24 results. Next

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

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