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 1-10 of 12 results. Next

A365178 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x).

Original entry on oeis.org

1, 1, 5, 30, 210, 1595, 12791, 106574, 913562, 8004861, 71375653, 645536234, 5907683486, 54605672300, 509043322720, 4780441915832, 45182744331388, 429472919087158, 4102806757542542, 39370967793387086, 379335734835510622, 3668220243145708341
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002294, A365180, A365181, A365182, A365183.
Cf. A364475, A365184.
Cf. A002293, A364987.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(k, n-k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002293(k).

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

Cf. A364984, A364985, A364986.
Cf. A006153, A295238, A364987.
Cf. A001764, A307678.

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

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

Cf. A336950, A381998, A381999, A382000, A382001.
Cf. A002293, A364987, A381983.

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

A365177 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 10, 201, 6220, 261465, 13925286, 898994383, 68240292856, 5956670911041, 587896878021130, 64738492669538391, 7869297152389747284, 1046629627952327990545, 151192146681811716344878, 23573456446401808474471455, 3945806733850334447131941616
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365175, A365176.

Programs

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

Formula

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

A377504 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 36, 735, 21972, 871995, 43308378, 2588123811, 180990517032, 14507325973395, 1311719669172750, 132102208441613883, 14666354372331521676, 1779817542971018697003, 234399632982398657764578, 33297612755940733707395955, 5075234637265322738651060688, 826215756199826873368252279971
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377528.
Cf. A006632, A364987.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+2, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+2,k)/( (k+1)*(n-k)! ).

A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

Original entry on oeis.org

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Crossrefs

Cf. A006153, A295238, A364983, A364987.
Cf. A002294.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.
a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024

A377576 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 52, 1116, 34408, 1394340, 70298424, 4248802516, 299752943200, 24196951718532, 2200519882434280, 222683725755611604, 24824104612186789584, 3023063956714780554628, 399343825987950226379416, 56879649386095684434783060, 8689968793295620150120679104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377575.
Cf. A364987, A377577.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+4, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+4,k)/( (k+1)*(n-k)! ).

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

Cf. A364987, A364989, A365176, A365177.
Cf. A161633, A213644, A364984.

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)! ).

A365176 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 10, 195, 5836, 236925, 12177966, 758458603, 55528414264, 4674208189977, 444823048027450, 47227542351423951, 5534636939373353604, 709653811287800826421, 98825110036657191358822, 14853654178825132742729715, 2396666529204491489278153456
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365175, A365177.

Programs

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

Formula

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

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 13, 217, 5937, 223641, 10725433, 625007993, 42883208609, 3386452550689, 302545287708201, 30170153462509545, 3322052185576104049, 400328811249634307249, 52406094009429908677049, 7405663486143907784247481, 1123601498350780798756198209, 182173718779147621454796872769
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2025

Keywords

Crossrefs

Cf. A381984, A381986.
Cf. A001764, A002293, A346646, A364987.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A364987. F(x) = B(x*A(x)) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x))^k/k ).
a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002293(k)/(n-k)!.
Showing 1-10 of 12 results. Next

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

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