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 1-6 of 6 results.

A365055 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x)^3 ).

Original entry on oeis.org

1, 1, 8, 121, 2818, 89006, 3559504, 172489948, 9825889532, 643567980808, 47654835126436, 3936868360416476, 358990055621209984, 35816155847478234424, 3880967272702222156952, 453886307361640406266456, 56985342864303337121933584, 7644651551838264804179619200
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2023

Keywords

Links

Crossrefs

Cf. A365053, A365054, A365056.
Cf. A363478.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x*(1+x/2))/3)))

Formula

E.g.f.: exp( -LambertW(-3*x * (1+x/2))/3 ).
a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (3*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!.

A378046 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^3 / (1+x)^2).

Original entry on oeis.org

1, 2, 11, 169, 4049, 132881, 5542495, 280694135, 16730578625, 1147444968385, 89015365063991, 7707022678811567, 736734708409976017, 77070404075178587633, 8757816984586841345231, 1074244834335107678837191, 141469329806979182825146625, 19908315372027482035799282177
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A377826, A378045.
Cf. A363478.

Programs

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

Formula

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

A378093 E.g.f. satisfies A(x) = exp( x * (1-x)^2 * A(x)^3 ) / (1-x).

Original entry on oeis.org

1, 2, 13, 187, 4421, 145381, 6106885, 312010217, 18775791529, 1300609323577, 101932831136801, 8917429459192717, 861423205666601869, 91071085791088039781, 10459294205668851438589, 1296711971347861868098561, 172604468588739615868724945, 24551969347625035312300681969
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2024

Keywords

Links

Crossrefs

Cf. A352410, A378092.
Cf. A363478.

Programs

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

Formula

E.g.f.: exp( -LambertW(-3*x/(1-x))/3 )/(1-x).
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n,k)/k!.
Showing 1-6 of 6 results.

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