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.

A380050 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x)*A(x) ).

Original entry on oeis.org

1, 1, 3, 9, 25, 25, -429, -4151, -8175, 320625, 5241475, 23329801, -705579159, -18521117303, -150119840493, 3366485315145, 138253031778721, 1780881865542625, -28047359274759549, -1854674541474191351, -34985197604145203655, 332608115115937927161
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A006153, A380051.
Cf. A297010, A380044.

Programs

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

Formula

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

A380047 E.g.f. A(x) satisfies A(x) = 1 + 3*x*exp(x)*A(x)^(1/3).

Original entry on oeis.org

1, 3, 12, 45, 132, 135, -702, 6573, 111576, -634581, -19482690, 104641713, 5438689380, -21226768017, -2173847986086, 3249084663765, 1168505502268848, 2167390942251219, -807540016560944778, -5035872168333504807, 693302551375611396540, 8209523136574257223383
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A006153, A380046.
Cf. A380051.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380051.
a(n) = n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(k/3+1,k)/( (k/3+1)*(n-k)! ).

A380094 E.g.f. A(x) satisfies A(x) = ( 1 + 3*x*exp(x*A(x)) )^(1/3).

Original entry on oeis.org

1, 1, 0, 7, -28, 405, -4514, 75313, -1336824, 28494793, -672782950, 17874984501, -521966931716, 16702822898749, -579928752836874, 21736834275178345, -874384126286848624, 37581186999500130321, -1718628399364227445070, 83327485224351815544925
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A161631, A380093.
Cf. A380051.

Programs

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

Formula

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

A380134 Expansion of e.g.f. (1 + 3*x*exp(x))^(1/3).

Original entry on oeis.org

1, 1, 0, 1, -4, 25, -194, 1813, -19816, 248113, -3502630, 55052701, -953576876, 18048491305, -370623627178, 8207063150245, -194950421191504, 4944881412682081, -133394451535683278, 3813510163227155245, -115170227064335439700, 3663942710200202043481
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A028310, A380133.
Cf. A380051, A380094.
Cf. A380017.

Programs

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

Formula

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

A380081 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + 3*x*exp(x))^(1/3) ).

Original entry on oeis.org

1, 1, 2, 7, 36, 245, 2086, 21357, 255704, 3507625, 54258570, 934600601, 17743468612, 368146983789, 8288468950958, 201258635444245, 5243025162331056, 145871455305823697, 4316920830720239122, 135408946029576741297, 4487574630295937337500, 156686063319198543135061
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A161633, A380080.
Cf. A380051.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = ( 1 + 3*x*A(x)*exp(x*A(x)) )^(1/3).
a(n) = (n!/(n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n/3+1/3,k)/(n-k)!.
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.