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.

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)!.

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)!.

A379879 E.g.f. A(x) satisfies A(x) = exp(-x) + x*A(x)^2.

Original entry on oeis.org

1, 0, 1, 5, 41, 439, 5869, 94275, 1770705, 38102255, 924580181, 24984120523, 744154938361, 24224671103463, 855748556756157, 32604902612628419, 1332864500919743393, 58192519232324179423, 2702582455278623736997, 133037424985668849756603
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A000166, A379878.
Cf. A377859, A379868.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(2*exp(-x)/(1+sqrt(1-4*x*exp(-x)))))
  • PARI
    a(n) = -n!*sum(k=0, n, (-k-1)^(n-k-1)*binomial(2*k, k)/(n-k)!);

Formula

E.g.f.: 2*exp(-x)/(1 + sqrt(1 - 4*x*exp(-x))).
a(n) = -n! * Sum_{k=0..n} (-k-1)^(n-k-1) * binomial(2*k,k)/(n-k)!.
a(n) ~ sqrt(1 + LambertW(-1/4)) * n^(n-1) / (2^(3/2) * (-LambertW(-1/4))^(n+1) * exp(n)). - Vaclav Kotesovec, Jan 23 2025

A379875 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)) + x.

Original entry on oeis.org

1, 0, 1, -4, 29, -256, 2797, -36352, 549145, -9468928, 183661721, -3960254464, 94011364405, -2436944723968, 68503370394565, -2075866971897856, 67464214813124273, -2340885649895194624, 86377064031382020913, -3377541983440381935616, 139515670016074334382541
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377859, A379876.

Programs

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

Formula

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

A377860 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(x) ).

Original entry on oeis.org

1, 1, 5, 44, 577, 10104, 222133, 5886880, 182775969, 6509571200, 261665344261, 11720054882304, 578878362625825, 31259890045425664, 1832295378792935925, 115862322601669627904, 7861907382202262095297, 569837358810005613281280, 43939338917141224534941829
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377859, A377861.
Cf. A377832.

Programs

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

Formula

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

A377861 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(x) ).

Original entry on oeis.org

1, 2, 15, 206, 4193, 113904, 3882511, 159475280, 7672503681, 423360926720, 26362968645071, 1829066086810368, 139929538526047585, 11703312997355442176, 1062423600515479191375, 104042389901715413633024, 10933256593926589800851969, 1227201235266954603172331520
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377859, A377860.
Cf. A377833.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-1)^k*(n+1)^(k-1)*binomial(4*n-k+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(-x * A(x))/(1 - x*A(x))^3.
a(n) = n! * Sum_{k=0..n} (-1)^k * (n+1)^(k-1) * binomial(4*n-k+2,n-k)/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)!.

A379937 E.g.f. A(x) satisfies A(x) = ( exp(-x*A(x)^(1/2)) + x*A(x) )^2.

Original entry on oeis.org

1, 0, 2, 4, 48, 328, 4240, 52092, 842240, 14598352, 294741504, 6501719860, 159434125312, 4248764847000, 123112522876928, 3840463241458732, 128576024097914880, 4594095412384753312, 174592522399006720000, 7030376888543624506212, 299062278252922180468736
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Crossrefs

Cf. A379866, A379932.
Cf. A377859.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(1-x)*exp(x))/x)^2))
  • PARI
    a(n) = -2*n!*sum(k=0, n, (-n-2)^(n-k-1)*binomial(n+k+1, k)/(n-k)!);

Formula

E.g.f.: ( (1/x) * Series_Reversion( x*(1-x)*exp(x) ) )^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377859.
a(n) = -2 * n! * Sum_{k=0..n} (-n-2)^(n-k-1) * binomial(n+k+1,k)/(n-k)!.
Showing 1-8 of 8 results.

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