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 1-7 of 7 results.

A379868 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x*A(x)^2.

Original entry on oeis.org

1, 0, 1, -1, 25, -101, 2281, -19895, 472305, -6760297, 177126121, -3578690435, 105341330953, -2743981145933, 91092111623241, -2888769295882111, 107832291781283809, -4009180998104138321, 167254334458983887689, -7105017992715364001147, 328862774630320838523321
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A108919, A379871.
Cf. A377859, A379879.
Cf. A364982, A367866.

Programs

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

Formula

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

A379877 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x*A(x)^3.

Original entry on oeis.org

1, 0, 1, 2, 33, 244, 4345, 61830, 1332961, 28087208, 739562481, 20380504330, 644853623425, 21767589641628, 810480865644073, 32246095869576974, 1385625666085792065, 63366863108725330000, 3090966367543869021409, 159607809547688836085778, 8718178798812199357657441
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Links

Crossrefs

Cf. A379871, A379876, A379878.

Programs

  • Mathematica
    Table[-n! * Sum[(-2*n - 1)^(n-k-1) * Binomial[2*n + k, k]/(n-k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2025 *)
  • PARI
    a(n) = -n!*sum(k=0, n, (-2*n-1)^(n-k-1)*binomial(2*n+k, k)/(n-k)!);

Formula

a(n) = -n! * Sum_{k=0..n} (-2*n-1)^(n-k-1) * binomial(2*n+k, k) / (n-k)!.
a(n) = U(-n, -3*n, -1 - 2*n)/(1 + 2*n), where U is the Kummer U function. - David Trimas, Jan 09 2025
a(n) ~ 2^(3*n) * n^(n-1) / (sqrt(3) * exp(2*n + 1/2)). - Vaclav Kotesovec, Jan 15 2025

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

A379858 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^3) + x.

Original entry on oeis.org

1, 0, 1, -10, 157, -3136, 77509, -2288896, 78824953, -3105906688, 137925180361, -6818997285376, 371578940493589, -22130352562929664, 1430368670554859533, -99722125119137591296, 7459992570265962997489, -596072767690463855509504, 50666927756525446827810961
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379871, A379910, A379911.
Cf. A379856, A379875.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-3*n+4*k-1)^(n-k)*binomial(3*n-3*k+1, k)/((3*n-3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (-3*n+4*k-1)^(n-k) * binomial(3*n-3*k+1,k)/( (3*n-3*k+1)*(n-k)! ).

A379910 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^3) + x*A(x).

Original entry on oeis.org

1, 0, 1, -7, 93, -1531, 32053, -805659, 23747545, -803011879, 30657419361, -1304526138895, 61227806142517, -3142500604364811, 175099735351517005, -10526856054032137891, 679212922630849128753, -46816385951481961302991, 3433289231599510254603193
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A379858, A379871, A379911.

Programs

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

Formula

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

A379911 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^3) + x*A(x)^2.

Original entry on oeis.org

1, 0, 1, -4, 53, -656, 11917, -244896, 6080265, -171274240, 5480682041, -195121452032, 7672945614589, -329902678161408, 15405361461450885, -776248476561903616, 41985495698339969681, -2426188309657908936704, 149180887282915274036977, -9725086440331395237937152
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377859, A379868, A379879.
Cf. A379858, A379871, A379910.

Programs

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

Formula

a(n) = -n! * Sum_{k=0..n} (-3*n+2*k-1)^(n-k-1) * binomial(3*n-k,k)/(n-k)!.
Showing 1-7 of 7 results.

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