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 23 results. Next

A088695 E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

Original entry on oeis.org

1, 1, 5, 40, 485, 7776, 156457, 3788800, 107414505, 3491200000, 128019454541, 5229222395904, 235490648957005, 11592449531084800, 619331166211640625, 35691050995648823296, 2206955604752999720273, 145757527499874820423680, 10240455593560436925898645
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2003

Keywords

Comments

Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Links

Crossrefs

Cf. A143768, A362771.
Cf. A365031, A365032.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)
  • PARI
    a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n),n,x)/(n+1)

Formula

a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).
a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - Vladimir Kruchinin, Aug 04 2011
a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - Vaclav Kotesovec, Jan 24 2014
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)) ). - Seiichi Manyama, Sep 23 2024

A377826 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 7, 49, 489, 6521, 108643, 2178107, 51084337, 1373054833, 41624314371, 1405311853595, 52299954524953, 2127347522554073, 93902399411048803, 4470613587492385051, 228362858274694209249, 12458393118650371672673, 722983769486947261178371
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377827, A377828.
Cf. A352410, A362771.

Programs

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

Formula

E.g.f.: (1+x) * exp( -LambertW(-x*(1+x)) ).
E.g.f.: -LambertW(-x*(1+x))/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k+1,n-k)/k!.
a(n) ~ sqrt(-2*sqrt(1 + 4*exp(-1)) + 2 + 8*exp(-1)) * 2^n * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(n+1) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 09 2024

A362773 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 79, 1377, 32161, 947623, 33746511, 1410518273, 67714577857, 3672410420871, 222082390164559, 14817864737168353, 1081393797641087841, 85691459902207874471, 7327398378967991154511, 672511583942513406768897, 65943097191889528063033729
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A047974, A362771.
Cf. A361065.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sqrt[LambertW[-2*x * (1+x)]/(-2*x * (1+x))], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 10 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*(1+x))/2)))

Formula

E.g.f.: exp( -LambertW(-2*x * (1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: sqrt(LambertW(-2*x * (1+x))/(-2*x * (1+x))).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * 2^(n-1) * n^(n-1) / ((-1 + sqrt(1 + 2*exp(-1)))^n * exp(n-1)). (End)

A363478 E.g.f. satisfies A(x) = exp(x * (1 + x) * A(x)^3).

Original entry on oeis.org

1, 1, 9, 142, 3481, 115476, 4849639, 246746662, 14756605329, 1014635520424, 78869009859751, 6839463570354306, 654661145565724345, 68559809182824171148, 7797979656027302949159, 957275139494698134599806, 126152927575064012671549729
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A362771, A362773.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x*(1+x))/3)))

Formula

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

A362772 E.g.f. satisfies A(x) = exp( x * (1+x)^2 * A(x) ).

Original entry on oeis.org

1, 1, 7, 58, 725, 11816, 239047, 5794972, 163861609, 5299694704, 193052158091, 7823764856084, 349236133422013, 17028109232138824, 900544754206010383, 51348494205747851116, 3140366001277974883793, 205067625446428300157408
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A000272, A362771.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x)^2))))

Formula

E.g.f.: exp( -LambertW(-x * (1+x)^2) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k,n-k)/k!.

A363355 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x)^2)).

Original entry on oeis.org

1, 1, 5, 46, 641, 11996, 282907, 8060242, 269429729, 10341367480, 448317429011, 21667926214694, 1155321200076529, 67370686916807236, 4265392644606677963, 291391173322695366106, 21365209437807863776193, 1673543873372595900318704
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Crossrefs

Cf. A088695, A362771, A363356.
Cf. A365012.

Programs

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

Formula

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

A365038 E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)).

Original entry on oeis.org

1, 1, 1, -2, 9, -44, 175, 246, -21007, 396712, -5576769, 57840850, -151112951, -14137899060, 539212013327, -13335393617714, 239914650459105, -1990873438067504, -76974185162417921, 5220283004540970282, -194958036625254566599, 5226632355735840377140
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2023

Keywords

Links

Crossrefs

Cf. A362771, A362773, A363478, A365039, A365040.
Cf. A361067.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(1+x)))))

Formula

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

A376145 E.g.f. satisfies A(x) = exp( x * (1+x)^3 * A(x) ).

Original entry on oeis.org

1, 1, 9, 88, 1265, 23916, 558427, 15608986, 508516017, 18936594712, 793902926771, 37017671474334, 1900666877186761, 106576903636156084, 6481047448001720427, 424870924596413523106, 29871349825140536394593, 2242231079099137007066544
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2024

Keywords

Links

Crossrefs

Cf. A362771, A362772, A376146.
Cf. A360076, A367789.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x)^3))))

Formula

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

A365039 E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)^2).

Original entry on oeis.org

1, 1, -1, 7, -79, 1201, -22961, 530167, -14372191, 447825889, -15776617249, 620209389031, -26918670325295, 1278598424153233, -65973615445792081, 3674793950748867031, -219773335672937703871, 14046128883828030510529, -955409650156763223984449
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2023

Keywords

Links

Crossrefs

Cf. A362771, A362773, A363478, A365038, A365040.
Cf. A361068.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x*(1+x))/2)))

Formula

E.g.f.: exp( LambertW(2*x * (1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (-2*k+1)^(k-1) * binomial(k,n-k)/k!.

A365040 E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)^3).

Original entry on oeis.org

1, 1, -3, 34, -623, 15636, -499277, 19382686, -886663647, 46716323752, -2786249779829, 185574001203834, -13652735530485647, 1099602989008154476, -96230900016000250269, 9092834662610587023286, -922622745817066477888703, 100054409045940667152740304
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2023

Keywords

Links

Crossrefs

Cf. A362771, A362773, A363478, A365038, A365039.
Cf. A361069.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x*(1+x))/3)))

Formula

E.g.f.: exp( LambertW(3*x * (1+x))/3 ).
a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(k,n-k)/k!.
Showing 1-10 of 23 results. Next

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

Content is available under The OEIS End-User License Agreement.