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.

A379871 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^3) + x*A(x)^3.

Original entry on oeis.org

1, 0, 1, -1, 37, -151, 5041, -45277, 1548457, -23466079, 857700181, -18904086037, 752753527021, -21985835786383, 961877988836857, -34996151990315341, 1686330291491184337, -73237182836313686719, 3882675760305075969949, -195288563442324161608165
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379876, A379877, A379878.
Cf. A108919, A379868.

Programs

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

Formula

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

A379876 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)) + x*A(x)^3.

Original entry on oeis.org

1, 0, 1, 5, 53, 689, 11509, 231083, 5448841, 147483665, 4508952641, 153682778435, 5778729641629, 237643665397241, 10610714800698349, 511207317411929339, 26434273616510818961, 1460296693254659368481, 85832214445015447832569, 5348490494660467991798003
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379871, A379877, A379878.
Cf. A377859, A379875.

Programs

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

Formula

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

A379878 E.g.f. A(x) satisfies A(x) = exp(-x) + x*A(x)^3.

Original entry on oeis.org

1, 0, 1, 8, 97, 1544, 30673, 732752, 20486401, 656713520, 23755416481, 957430990328, 42552022022497, 2067669370359800, 109058922249721585, 6205740584180119424, 378947624701223801089, 24718152376534891564256, 1715322065909959400535361, 126186162087426817989206888
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379871, A379876, A379877.
Cf. A000166, A379879.

Programs

  • Mathematica
    Table[-n! * Sum[(-2*k-1)^(n-k-1) * Binomial[3*k, k] / (n-k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *)
  • PARI
    a(n) = -n!*sum(k=0, n, (-2*k-1)^(n-k-1)*binomial(3*k, k)/(n-k)!);

Formula

a(n) = -n! * Sum_{k=0..n} (-2*k-1)^(n-k-1) * binomial(3*k,k)/(n-k)!.
a(n) ~ (-1)^n * sqrt(-LambertW(-8/27) - 1) * 2^n * n^(n-1) / (3 * exp(n) * LambertW(-8/27)^(n + 1/2)). - Vaclav Kotesovec, Jan 23 2025

A379856 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x.

Original entry on oeis.org

1, 0, 1, -7, 81, -1181, 21373, -462267, 11663137, -336711385, 10955316501, -396815693759, 15840688752529, -691086583866069, 32717602050027469, -1670649590632148611, 91530694441643402817, -5355984871255569700913, 333392838283336197688741
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379868, A379877, A379909.
Cf. A379858, A379875.

Programs

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

Formula

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

A379909 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x*A(x).

Original entry on oeis.org

1, 0, 1, -4, 41, -456, 6817, -120044, 2497105, -59445136, 1599030881, -47923901268, 1584315183673, -57269439049304, 2247345360390145, -95147690776024636, 4323183446836151201, -209835113176652954400, 10835768876261196612673, -593183387438256595660964
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379856, A379868, A379877.

Programs

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

Formula

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

A379939 E.g.f. A(x) satisfies A(x) = ( exp(-x*A(x)^(2/3)) + x*A(x) )^3.

Original entry on oeis.org

1, 0, 3, 6, 117, 852, 16335, 231354, 5169801, 109149768, 2929053339, 81073827150, 2593779841917, 87970941597276, 3298932148606887, 131818125152516418, 5692856683050644625, 261303806858004143376, 12794600152539073620531, 662722202747157809616918
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A379877, A379937.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A379877.
a(n) = -3 * n! * Sum_{k=0..n} (-2*n-3)^(n-k-1) * binomial(2*n+k+2,k)/(n-k)!.
Showing 1-6 of 6 results.

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