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.

A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

Programs

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

Formula

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*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

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

A380513 Expansion of e.g.f. exp(x*G(x)) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 31, 649, 20241, 831691, 42281023, 2558247441, 179401012129, 14301145772371, 1276863732880671, 126200478678828313, 13677209933635675441, 1612657716714084149019, 205505541279096688937791, 28144314031348292162103841, 4122178445898981809990411073, 642961375302043479923591655331
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A002293, A380514, A380515, A380516.
Cf. A000262, A080893, A251569.

Programs

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

Formula

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

A380514 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 5, 67, 1537, 50021, 2107021, 108885295, 6665443457, 471522589417, 37843890892021, 3397250515809371, 337267132243022785, 36687625652474612557, 4339368321317331858557, 554467482301151809302151, 76112537023512618262963201, 11170667360636927554290623825, 1745500813880455301486766050917
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A002293, A069271, A380513, A380515, A380516.

Programs

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

Formula

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

A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 43, 73, 1, 1, 1, 9, 91, 529, 501, 1, 1, 1, 11, 157, 1753, 8501, 4051, 1, 1, 1, 13, 241, 4129, 45001, 169021, 37633, 1, 1, 1, 15, 343, 8041, 146001, 1447471, 4010455, 394353, 1, 1, 1, 17, 463, 13873, 362501, 6502681, 56041987, 110676833, 4596553, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,      1,      1, ...
  1,   1,    1,     1,      1,      1, ...
  1,   3,    5,     7,      9,     11, ...
  1,  13,   43,    91,    157,    241, ...
  1,  73,  529,  1753,   4129,   8041, ...
  1, 501, 8501, 45001, 146001, 362501, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000262, A251568, A380512, A380516.
Cf. A355262, A382100.

Programs

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

Formula

A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).

A380638 Expansion of e.g.f. exp(x*G(4*x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 33, 2209, 226753, 31555521, 5557183201, 1185423664993, 297171500140929, 85638231765516673, 27896677183469054881, 10137203757416219332641, 4065668625283435566910273, 1783936343221839549449049409, 850091650335726912762794748513, 437197222292805469886634467693281
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2025

Keywords

Links

Crossrefs

Cf. A002293, A380516.

Programs

  • PARI
    a(n) = if(n==0, 1, 4^(n-1)*(n-1)!*pollaguerre(n-1, 3*n+1, -1/4));

Formula

E.g.f.: exp( (G(4*x)-1)/4 ), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 4^k * binomial(4*n,k)/(n-k-1)! for n > 0.
a(n+1) = 4^n * n! * LaguerreL(n, 3*n+4, -1/4).
a(n) ~ 2^(10*n - 1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 1/12)). - Vaclav Kotesovec, Jan 29 2025
a(n) = (-4)^(n-1)*U(1-n, 2+3*n, -1/4), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.