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 1-5 of 5 results.

A380051 E.g.f. A(x) satisfies A(x) = ( 1 + 3*x*exp(x)*A(x) )^(1/3).

Original entry on oeis.org

1, 1, 2, 1, -12, -15, 526, 1617, -49608, -302111, 8126010, 85724001, -2020009628, -34232466255, 696686324166, 18267485751985, -310973114236944, -12533263924965183, 168118610439268594, 10727427541319225793, -100693940482485604260, -11178369799980253348079
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A006153, A380050.
Cf. A380047.

Programs

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

Formula

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

A380093 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x*A(x)) ).

Original entry on oeis.org

1, 1, 1, 6, 13, 180, 501, 13720, 34777, 2014992, 2512585, 491642976, -564313947, 181714012480, -836832558275, 95473740036480, -856984734161999, 68029327826567424, -954950936641491951, 63368301861354866176, -1238053892876418633155, 74904417332353810338816
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A161631, A380094.
Cf. A380035, A380050.

Programs

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

Formula

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

A380046 E.g.f. A(x) satisfies A(x) = 1 + 2*x*exp(x)*A(x)^(1/2).

Original entry on oeis.org

1, 2, 8, 36, 176, 840, 3312, 4592, -85888, -893664, 1375040, 165097152, 2297399040, -437916544, -676590342400, -13778476089600, -35262701498368, 5528190100333056, 159800245551129600, 1036568296401259520, -77532370748157030400, -3135837171024874272768
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A006153, A380047.
Cf. A380050.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380050.
E.g.f.: exp( 2*arcsinh(x*exp(x)) ).
E.g.f.: ( x*exp(x) + sqrt(1 + x^2*exp(2*x)) )^2.
a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(k/2+1,k)/( (k/2+1)*(n-k)! ).

A380080 Expansion of e.g.f. (1/x) * Series_Reversion( x / sqrt(1 + 2*x*exp(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 109, 1045, 12501, 179599, 3015657, 57988809, 1257058585, 30337358491, 806837271021, 23448335293981, 739379851041573, 25143044445680295, 917252832237053521, 35735484803144976145, 1480838869407287923569, 65038486139094829172275, 3017945328547452509505045
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A161633, A380081.
Cf. A380050.

Programs

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

Formula

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

A380133 Expansion of e.g.f. sqrt(1 + 2*x*exp(x)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, -9, 70, -335, 504, 11935, -182094, 1525833, -4911764, -99495473, 2430329070, -29988416159, 158542630224, 2868272912511, -102775471991126, 1714422613948345, -13166449628575404, -209400601689898289, 10598981162761786950, -227206614609529433199
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A028310, A380134.
Cf. A380050, A380093.
Cf. A380015.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(1/2,k)/(n-k)!.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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