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

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)!.

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

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

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

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)! ).

A379866 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + x)^2 ).

Original entry on oeis.org

1, 0, 2, -2, 56, -222, 5332, -45782, 1127408, -15972542, 428055644, -8598013734, 256717806952, -6667767637598, 223389539254676, -7076616268104278, 265762684840216544, -9880557234248622462, 413902270494309471436, -17591536945041528005318, 816621849842712202724696
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (exp(-x*A(x)) + x*A(x))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A379868.
a(n) = -2 * n! * Sum_{k=0..n} (-2*n+k-2)^(n-k-1) * binomial(2*n+1,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

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)!.

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

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)!.
Showing 1-6 of 6 results.