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 1-8 of 8 results.

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!.

A363358 E.g.f. satisfies A(x) = exp(x * A(x)^2 * (1 + x * A(x)^2)).

Original entry on oeis.org

1, 1, 7, 91, 1809, 48521, 1643863, 67381875, 3243606817, 179405231761, 11213025902631, 781604862035339, 60120379931640625, 5058593367221610009, 462199816484860893559, 45574025454771003821731, 4823543138131670132557377, 545448517762149418525390625
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Crossrefs

Cf. A362773, A363357.
Cf. A088695, A363479.
Cf. A361093.

Programs

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

Formula

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

A365054 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 64, 1038, 22666, 624448, 20801628, 813473468, 36543076444, 1854702411336, 104970490358944, 6555275229438664, 447773277245296536, 33211911279540910400, 2658266282912883209296, 228375288313274403201552, 20961681963345040127314192
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2023

Keywords

Links

Crossrefs

Cf. A365053, A365055, A365056.
Cf. A362474, A362773.

Programs

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

Formula

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

A363357 E.g.f. satisfies A(x) = exp(x * A(x)^2 * (1 + x * A(x))).

Original entry on oeis.org

1, 1, 7, 85, 1581, 39501, 1244953, 47426373, 2120506489, 108894505753, 6317267871501, 408637512353049, 29164082035045477, 2276557391070945477, 192956160476285907457, 17647873882378895267821, 1732445579330211460781937, 181694902682241512454842673
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Crossrefs

Cf. A362773, A363358.
Cf. A361142.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (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!.

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!.

A378045 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^2 / (1+x)).

Original entry on oeis.org

1, 2, 9, 100, 1693, 39046, 1140589, 40379872, 1680490361, 80409242314, 4349556199441, 262478904794140, 17482853419143061, 1274026039224276430, 100830973069183104245, 8612770277501109271576, 789749958006001265241073, 77375794118912255978104978, 8066966112797470401673208089
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A377826, A378046.
Cf. A377894, A378043.
Cf. A362773.

Programs

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

Formula

E.g.f.: (1+x) * exp( -LambertW(-2*x*(1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k+1,n-k)/k!.
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * (1 + sqrt(1 + 2*exp(-1))) * 2^(n-2) * n^(n-1) / ((sqrt(1 + 2*exp(-1)) - 1)^n * exp(n-1)). - Vaclav Kotesovec, Nov 15 2024
Showing 1-8 of 8 results.

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

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