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 1-3 of 3 results.

A380039 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)) )^(1/3).

Original entry on oeis.org

1, 1, 6, 61, 908, 17865, 438286, 12901735, 443475432, 17443879057, 773018191610, 38117147134671, 2070381313048588, 122841147634754185, 7905667340470592070, 548555101319868261655, 40825552788531622527056, 3244188226183716688784289, 274164589130871765969460594
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Programs

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

Formula

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

A380043 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)^3) )^(1/3).

Original entry on oeis.org

1, 1, 6, 73, 1364, 34585, 1110406, 43200535, 1975744856, 103892750209, 6176282882570, 409635957376591, 29988473838531748, 2402004132488328433, 208956515057627326094, 19619264794744128427495, 1977503574407863125008816, 212975277029523353673126529, 24408338689788753822318157330
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x/(1+3*x*exp(x)))/x)^(1/3)))
    
  • PARI
    a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n+1/3, k)/(n-k)!)/(3*n+1);

Formula

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

A380040 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)) )^(2/3).

Original entry on oeis.org

1, 2, 14, 170, 3000, 69930, 2033212, 70972734, 2894590064, 135164076722, 7113787010100, 416759006663142, 26903080612468744, 1897553477118350922, 145204649027247413996, 11982094054396851014030, 1060673494236770414806752, 100265097180082772515691874, 10080871201186661027182272868
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380041.
a(n) = 2 * n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(2*n/3+k/3+2/3,k)/( (2*n+k+2)*(n-k)! ).
Showing 1-3 of 3 results.