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.

A373519 Expansion of e.g.f. exp(x/(1 - x^4)^(1/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 31, 181, 631, 1681, 60481, 687961, 4379761, 19982161, 802740511, 13848694861, 131732390791, 873339798241, 38385869907841, 894783905472241, 11506538747852641, 101612306808695521, 4824806928717603871, 142148609212891008421
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Links

Crossrefs

Cf. A293507, A373520, A373521.
Cf. A373524.
Cf. A000262, A012150, A373517.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[E^(x/(1 - x^4)^(1/4)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/4-1, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/4-1,k)/(n-4*k)!.
a(n) == 1 mod 30.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (5*n^4 - 80*n^3 + 505*n^2 - 1480*n + 1681)*a(n-4) - 5*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(2*n^2 - 24*n + 85)*a(n-8) + 5*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*(2*n^2 - 32*n + 135)*a(n-12) - 5*(n-16)*(n-15)*(n-14)*(n-13)*(n-12)^2*(n-11)*(n-10)^2*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*a(n-16) + (n-20)*(n-19)*(n-18)*(n-17)*(n-16)^2*(n-15)*(n-14)*(n-13)*(n-12)^2*(n-11)*(n-10)*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*a(n-20).
a(n) ~ 5^(-1/2) * exp(5*n^(1/5)/4 - n) * n^(n - 2/5).
(End)

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)

A373522 Expansion of e.g.f. exp(x * (1 + x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 9, 41, 121, -279, -1679, 1009, 259281, 1173041, 669241, -267141159, -1295686391, 10821721, 650092657761, 3480768830561, 17723446561, -2911516748764191, -17068971040559639, 427036022281, 21673592659354854681, 137752098937383025481
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190875, A373523.
Cf. A373517.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n/3-k,k)/(n-3*k)!.
a(n) == 1 mod 8.

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).

A373539 Expansion of e.g.f. exp(x/(1 + x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, -7, -39, -119, 841, 10641, 59473, -393679, -9119439, -77841719, 453247081, 17769103353, 210702481081, -1002688100959, -65813075987679, -1022777654395679, 3554736409105633, 413233827275657241, 8091508938651283321, -16214426267734966039
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2024

Keywords

Crossrefs

Cf. A012019, A373540.
Cf. A373517.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n/3-1,k)/(n-3*k)!.
a(n) == 1 mod 8.

A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 73, 361, 1081, 93241, 912241, 4907953, 476295121, 7244922961, 58360393081, 6211842488281, 130899060524233, 1435239754046281, 164948740478252641, 4498516738183799521, 63300797606830713121, 7772118657831401082913, 262261735708117281036841
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A189054, A371458, A373517.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1 - 9*x^3)^(1/3)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    nmax = 20; CoefficientList[Series[E^(x/(1 - 9*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, 9^k*binomial(n/3-1, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k)/(n-3*k)!.
a(n) == 1 mod 72.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (36*n^3 - 324*n^2 + 1008*n - 1079)*a(n-3) - 162*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 2916*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - 6561*(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) ~ 3^(2*n/3 - 1/4) * exp(4*3^(-3/2)*n^(1/4) - n) * n^(n - 3/8) / 2. (End)
Showing 1-6 of 6 results.

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