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 1-7 of 7 results.

A364983 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^3.

Original entry on oeis.org

1, 1, 8, 111, 2332, 66125, 2368086, 102616759, 5222638856, 305436798009, 20186656927210, 1488021110087171, 121044207712073196, 10771321471267219525, 1040877104088653696606, 108549742436141933697135, 12151467262433697322437136, 1453367472748861203540942065
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.
a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - Vaclav Kotesovec, Nov 11 2024

A364985 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 8, 123, 2884, 91445, 3664926, 177796759, 10132646840, 663644108169, 49123993335130, 4055804550134051, 369544757016476196, 36834870020525413213, 3987179241476814768854, 465777171342934543710255, 58407238852473276959363056, 7825395596421876706944643985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Programs

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

Formula

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

A364986 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 8, 129, 3196, 107465, 4575966, 236120059, 14322901832, 998966928897, 78770826493210, 6929685905371691, 672900446143476156, 71491442785783506577, 8249400210035835040022, 1027394346436911560475915, 137360293432089585554830096
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Programs

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

Formula

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

A381175 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cos(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 69, 1224, 29465, 898320, 33187133, 1441200768, 71956238769, 4061414246400, 255737764687669, 17773804761259008, 1351494159065894857, 111608708333568036864, 9947544079380663728685, 951770403836914402099200, 97301151510219112917218657, 10585077723403580668983902208
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
    a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A365175 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)).

Original entry on oeis.org

1, 1, 10, 189, 5476, 215145, 10701006, 644909503, 45687408712, 3721382812305, 342689189598010, 35206864089944151, 3992473080042706524, 495361299387667990537, 66752437447119717428422, 9708649781691227748131535, 1515863453268825963300368656
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Programs

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

Formula

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

A381172 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cosh(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 75, 1416, 36065, 1160400, 45182347, 2066343552, 108594342369, 6449557524480, 427226389872491, 31230489190382592, 2497416890105693569, 216875134620623990784, 20324880119519860657515, 2044641793664946681446400, 219762483007148574205773377, 25134006030221243013604835328
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
    a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * A185951(n,k).

A382043 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^3*exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 10, 168, 4280, 146840, 6354432, 332467072, 20419261312, 1440559380096, 114820434103040, 10205253450850304, 1000815286620229632, 107355373421379825664, 12504295470535952613376, 1571670041412254073323520, 212035122185327799251468288, 30561822671438790519426154496
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * binomial(n+2*k+1,k)/((n+2*k+1) * (n-k)!).
Showing 1-7 of 7 results.