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 1-4 of 4 results.

A381997 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^4.

Original entry on oeis.org

1, 1, 12, 240, 7328, 303400, 15904032, 1010252320, 75442821120, 6478112692224, 628915387166720, 68121797696449024, 8144844724723482624, 1065508614975814537216, 151392999512027274215424, 23217165210450099377479680, 3822334349865128121165283328, 672407573328393115218009063424
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Programs

  • Maple
    A381997 := proc(n)
            n!*add((2*k)^(n-k)*binomial(4*k+1,k)/(4*k+1)/(n-k)!,k=0..n) ;
    end proc:
    seq(A381997(n),n=0..60) ;  # R. J. Mathar, Mar 12 2025
  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002293(k)/(n-k)!.
a(n) ~ 2^(n+1) * n^(n-1) * sqrt(1 + LambertW(27/128)) / (3^(3/2) * exp(n) * LambertW(27/128)^n). - Vaclav Kotesovec, Mar 22 2025

A382000 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^5.

Original entry on oeis.org

1, 1, 14, 342, 12872, 659280, 42828912, 3375009568, 312860626304, 33361836534144, 4023352486200320, 541461682626399744, 80448618080927609856, 13079749459734097573888, 2309915877337042992324608, 440332184936376095626076160, 90117169223076699520606896128
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002294(k)/(n-k)!.
a(n) ~ 2^(n-3) * n^(n-1) * sqrt(5*(1 + LambertW(512/3125))) / (exp(n) * LambertW(512/3125)^n). - Vaclav Kotesovec, Mar 22 2025

A382001 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^6.

Original entry on oeis.org

1, 1, 16, 462, 20672, 1261400, 97728672, 9190016416, 1016963389696, 129485497897728, 18648682990461440, 2997567408967391744, 531985786683988512768, 103321584851593487961088, 21798243872991807130685440, 4964302861788729054456729600, 1213816740632458735310221672448
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Comments

In general, if k>1 and e.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^k, then a(n) ~ sqrt(k) * sqrt(1 + LambertW(2*(k-1)^(k-1)/k^k)) * 2^n * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW(2*(k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Mar 22 2025

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002295(k)/(n-k)!.
a(n) ~ sqrt(3*(1 + LambertW(3125/23328))) * 2^(n + 1/2) * n^(n-1) / (5^(3/2) * exp(n) * LambertW(3125/23328)^n). - Vaclav Kotesovec, Mar 22 2025

A381998 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^2.

Original entry on oeis.org

1, 1, 8, 90, 1472, 31920, 865152, 28197904, 1075122176, 46976064768, 2315080816640, 127068467480064, 7688296957870080, 508450036968779776, 36490818871396499456, 2824787199565881477120, 234622076533699738861568, 20813348299168251651883008, 1964063064959266899440959488
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A000108(k)/(n-k)!.
From Vaclav Kotesovec, Mar 22 2025: (Start)
E.g.f.: 2/(1 + sqrt(1 - 4*exp(2*x)*x)).
a(n) ~ sqrt(1 + LambertW(1/2)) * 2^(n + 1/2) * n^(n-1) / (exp(n) * LambertW(1/2)^n). (End)
Showing 1-4 of 4 results.