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 11 results. Next

A293507 Expansion of e.g.f. exp(x/(1 - x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 378001, 5473441, 39972241, 199679041, 7005552841, 176899522801, 2186722497961, 17454339826561, 459473703430561, 16503993702423361, 306140370496394401, 3555223271216311681, 80917223353652470681, 3568770455830785208081
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2017

Keywords

Crossrefs

E.g.f.: exp(x/(1 - x^m)): A000262 (m=1), A088009 (m=2), A293493 (m=3), this sequence (m=4).
Cf. A293526.

Programs

  • Mathematica
    CoefficientList[Series[E^(x/(1 - x^4)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 11 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1-x^4))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)))))

Formula

E.g.f.: Product_{k>0} exp(x^(4*k-3)).
a(n) ~ exp(1/4 + sqrt(n) - n) * n^(n-1/4) / 2. - Vaclav Kotesovec, Oct 11 2017
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k+1)! * a(n-4*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k)/(n-4*k)!. - Seiichi Manyama, Jun 08 2024

A373517 Expansion of e.g.f. exp(x/(1 - x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 9, 41, 121, 1401, 11761, 61489, 864081, 10597841, 81833401, 1350154521, 21715461769, 225232218121, 4267472824161, 84597818284001, 1111699778741281, 23801969674626849, 558853937533757161, 8943028907965939081, 213696639293901810201
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A012150, A293493, A373518.
Cf. A373522.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n/3-1, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n/3-1,k)/(n-3*k)!.
a(n) == 1 mod 8.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (4*n^3 - 36*n^2 + 112*n - 119)*a(n-3) - 2*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 4*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - (n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).
a(n) ~ (1/2) * exp(4*n^(1/4)/3 - n) * n^(n - 3/8) * (1 - 35/(96*n^(1/4)) - 4367/(18432*sqrt(n)) + 1737829/(5308416*n^(3/4))). (End)

A293494 E.g.f.: exp(x^2/(1 - x^3)).

Original entry on oeis.org

1, 0, 2, 0, 12, 120, 120, 5040, 42000, 181440, 5473440, 46569600, 479666880, 12713500800, 123519876480, 2190354566400, 54922842374400, 652587628492800, 16192687807296000, 400439258111692800, 5994400896144460800, 182793304684134144000, 4590673238662192281600
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2017

Keywords

Crossrefs

Cf. A293493.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^2/(1-x^3))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-1)))))

Formula

E.g.f.: Product_{k>0} exp(x^(3*k-1)).
a(n) ~ exp(2*sqrt(3*n)/3 - n - 1/6) * n^(n-1/4) / (sqrt(2) * 3^(1/4)). - Vaclav Kotesovec, Oct 10 2017

A351934 Expansion of e.g.f. exp(x / (1 - x^3/6)).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 281, 1961, 10585, 59641, 509521, 4086061, 29792621, 288051765, 2997900361, 27857436881, 302238265841, 3759751338481, 43054257392545, 526471319508181, 7463835257515141, 101755979383832621, 1404876681654178041, 22115830210157954425
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000262, A351933, A351935, A351936.
Cf. A293493.

Programs

  • Mathematica
    m = 24; Range[0, m]! * CoefficientList[Series[Exp[x/(1 - x^3/6)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^3/6))))
  • PARI
    a(n) = if(n<4, 1, sum(k=0, (n-1)\3, (3*k+1)!/6^k*binomial(n-1, 3*k)*a(n-1-3*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1)!/6^k * binomial(n-1,3*k) * a(n-1-3*k) for n > 3.
a(n) ~ n^(n - 1/4) / (2^(n/3 + 5/12) * 3^(n/3 + 1/6) * exp(n - 2^(7/6) * 3^(-1/3) * sqrt(n) - 6^(-2/3))). - Vaclav Kotesovec, Mar 03 2022
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(6^k * (n-3*k)!). - Seiichi Manyama, Jun 08 2024

A373518 Expansion of e.g.f. exp(x/(1 - x^3)^(2/3)).

Original entry on oeis.org

1, 1, 1, 1, 17, 81, 241, 3361, 32481, 183457, 2534561, 36903681, 325995121, 4808334961, 90981786897, 1126128625441, 18354227120321, 415821040873281, 6714588707173441, 122710186163310337, 3174234862391072721, 63597591858999638161, 1308604168710672673841
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A293493, A373517.
Cf. A373523.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(2/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(2*n/3-k-1, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n/3-k-1,k)/(n-3*k)!.
a(n) == 1 mod 16.
From Vaclav Kotesovec, Sep 03 2025: (Start)
Recurrence: (n-9)*(2*n - 21)*a(n) = (2*n - 21)*(5*n^4 - 90*n^3 + 515*n^2 - 1229*n + 1071)*a(n-3) - (n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 570*n^3 + 5590*n^2 - 21846*n + 30303)*a(n-6) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 630*n^3 + 6940*n^2 - 31104*n + 49113)*a(n-9) - (n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 9)*(5*n^4 - 210*n^3 + 3215*n^2 - 20971*n + 47886)*a(n-12) + (n-15)*(n-14)^2*(n - 13)^2*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*(2*n - 9)*a(n-15).
a(n) ~ 2^(3/10) * 5^(-1/2) * exp(5*2^(-2/5)*n^(2/5)/3 - n) * n^(n - 3/10).
(End)

A293565 Expansion of e.g.f. Product_{k>=0} exp(-x^(3*k+1)).

Original entry on oeis.org

1, -1, 1, -1, -23, 119, -359, -4201, 58801, -359857, -2121839, 74851919, -784376999, -1072414201, 192596540137, -3167640042841, 9241215940321, 860635956717599, -21290802661288799, 157132294452113183, 5980504635868174921, -220612220726623344361
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Links

Crossrefs

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), this sequence (k=3), A293566 (k=4).
Cf. A293493.

Programs

  • Mathematica
    CoefficientList[Series[E^(x/(x^3 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^3-1))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(3*m+1)))))

Formula

E.g.f.: exp(x/(x^3 - 1)).
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^(n-k) * binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

A351905 Expansion of e.g.f. exp(x * (1 - x^3)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 18481, 178417, 902161, 3318481, -69866279, -1011908039, -7204341143, -36194591159, 726745175521, 14326789219681, 131901636673441, 840736509931297, -16060449291985079, -408041402342457239, -4618341644958693959, -35691963052019431079
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351906.
Cf. A190875, A293493.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^3))))
  • PARI
    a(n) = n!*sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)-4!*binomial(n-1, 3)*a(n-4));

Formula

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

A373680 Expansion of e.g.f. exp(x / (1 - x^3)^2).

Original entry on oeis.org

1, 1, 1, 1, 49, 241, 721, 16801, 204961, 1276129, 19968481, 417479041, 4522597201, 62399971921, 1685741065009, 28122880050721, 415551065616961, 12085752936331201, 281057646411506881, 4923299166925874689, 143004800073025326961, 4244797186148550210481
Offset: 0

Views

Author

Seiichi Manyama, Jun 13 2024

Keywords

Crossrefs

Cf. A293493, A373517, A373518.
Cf. A373654, A373685.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(2*n-5*k-1, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n-5*k-1,k)/(n-3*k)!.
a(n) == 1 (mod 48).

A373684 Expansion of e.g.f. exp(x / (1 - x^3)) / (1 - x^3).

Original entry on oeis.org

1, 1, 1, 7, 49, 181, 1201, 16171, 122977, 971209, 16788961, 206578351, 2094314641, 40260673597, 694617709969, 9088535091091, 187060085419201, 4109646055718161, 68029103349383617, 1498126927153125079, 39199612737165358321, 799248202624341298501
Offset: 0

Views

Author

Seiichi Manyama, Jun 13 2024

Keywords

Crossrefs

Cf. A002720, A373683.
Cf. A293493.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-3*k)!.
a(n) == 1 (mod 6).

A376563 E.g.f. satisfies A(x) = exp( x*A(x) / (1 - x^3*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 16, 149, 2016, 34447, 697264, 16349769, 438688000, 13302715931, 450127632384, 16805250091357, 685963808029696, 30389466219654375, 1452343581621354496, 74484586717220845073, 4080558161067766972416, 237830597688699873580339, 14694357436059217272832000
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2024

Keywords

Links

Crossrefs

Cf. A052873, A376558.
Cf. A293493, A376475.

Programs

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

Formula

E.g.f.: (1/x) * Series_Reversion( x*exp(-x / (1 - x^3)) ).
a(n) = n! * Sum_{k=0..floor(n/3)} (n+1)^(n-3*k-1) * binomial(n-2*k-1,k)/(n-3*k)!.
Showing 1-10 of 11 results. Next

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

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