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 1-6 of 6 results.

A380512 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 91, 1753, 45001, 1447471, 56041987, 2539200721, 131859347473, 7723214721271, 503787793244011, 36223369111466857, 2846582772323685721, 242741539845295265503, 22325483241906758894611, 2202979676409063904473121, 232158319570869255177386017, 26024052774273208806612761191
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Links

Crossrefs

Cf. A001764, A251569, A380511.
Cf. A000262, A251568, A380516.
Cf. A382033.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 2*n+1, -1));

Formula

E.g.f.: exp(G(x)-1), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} binomial(3*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 2*n+3, -1).
a(n) = (-1)^(n+1)*U(1-n, 2*(1+n), -1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x/(1+x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A382032 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x))^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 55, 937, 21741, 639841, 22839139, 958882289, 46304377849, 2528571710881, 154076164781991, 10364272238514217, 762867688235619877, 60989719558159065857, 5263030218009265964011, 487578723768665716788961, 48266847740986728218648433, 5084697384633390178057209793
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161630, A382033, A382034.
Cf. A000108, A377553, A382036.

Programs

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

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377553. F(x) = log(A(x))/x = C(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^2 ) ).

A382034 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 181, 5713, 246881, 13570081, 906180997, 71250724833, 6448375469665, 660286026034561, 75472025139452261, 9525947428687403473, 1315935073971181422721, 197485196722573989608289, 31993978774204625549549221, 5565216938342017912128576961, 1034506012356981473110554574145
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161630, A382032, A382033.
Cf. A002293, A377630.

Programs

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

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377630. F(x) = log(A(x))/x = B(x*A(x))^4.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^4 ) ).

A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 160, 4325, 157896, 7280077, 406085632, 26599741065, 2001864880000, 170236619802161, 16144762562002944, 1689534516295056301, 193403842876754728960, 24040636567791329323125, 3224829927677539092791296, 464325325579881390473331473, 71428455280041816247241637888
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A052873, A382036, A382038.
Cf. A001764, A377830, A382033.

Programs

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

Formula

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

A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

Programs

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

Formula

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.

A382059 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A161629, A382058.
Cf. A364938, A382033.
Cf. A002293, A377548, A382034.

Programs

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

Formula

Let F(x) be the e.g.f. of A377548. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^3 ) ).
a(n) = 3 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
Showing 1-6 of 6 results.

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