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 1-7 of 7 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)

A351906 Expansion of e.g.f. exp(x * (1 - x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, -15119, 1784161, 19902961, 119655361, 518763961, 1815974161, -212497445159, -3472602456959, -29605333299359, -177764320560959, -844590032480159, 97992221659873921, 2116963290135836521, 23379513665735470321
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351905.
Cf. A190877, A293507.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^4))))
  • PARI
    a(n) = n!*sum(k=0, n\5, (-1)^k*binomial(n-4*k, k)/(n-4*k)!);
  • PARI
    a(n) = if(n<5, 1, a(n-1)-5!*binomial(n-1, 4)*a(n-5));

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) - 5! * binomial(n-1,4) * a(n-5) for n > 4.

A373520 Expansion of e.g.f. exp(x/(1 - x^4)^(1/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 143641, 1829521, 12501721, 59922721, 2173048021, 44315751481, 478799701381, 3492321094081, 116722067432881, 3290135175240481, 50242015215929521, 508061488330088641, 16418736123292904941, 585427887134915295241
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A293507, A373519, A373521.
Cf. A373525.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[E^(x/(1 - x^4)^(1/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/2-k-1, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/2-k-1,k)/(n-4*k)!.
a(n) == 1 mod 60.
From Vaclav Kotesovec, Sep 03 2025: (Start)
Recurrence: (n-8)*a(n) = (n-8)*a(n-2) + 3*(n-4)*(n-3)*(n-2)*(n^2 - 11*n + 20)*a(n-4) + 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*a(n-6) - 3*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n^2 - 13*n + 32)*a(n-8) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-10) + (n-12)*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-12).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3).
(End)

A373521 Expansion of e.g.f. exp(x/(1 - x^4)^(3/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 91, 541, 1891, 5041, 249481, 3424681, 24365881, 119821681, 4208219731, 96849813061, 1133147785771, 8728726799521, 251218306095121, 8116398738418321, 138787857114672241, 1523943014238675361, 39648007379230971211, 1599866285860593980461
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A293507, A373519, A373520.
Cf. A373526.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(3*n/4-2*k-1, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(3*n/4-2*k-1,k)/(n-4*k)!.
a(n) == 1 mod 90.

A293526 E.g.f.: exp(x^3/(1 - x^4)).

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 360, 5040, 0, 60480, 3628800, 39916800, 19958400, 3113510400, 130767436800, 1318571654400, 3487131648000, 355687428096000, 12813639597158400, 126713646259200000, 1013709170073600000, 85161707377883136000, 2819368492175499264000
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2017

Keywords

Crossrefs

E.g.f.: Product_{k>0} exp(x^(m*k-1)): A088009 (m=2), A293494 (m=3), this sequence (m=4).
Cf. A293507.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^3/(1-x^4))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-1)))))

Formula

E.g.f.: Product_{k>0} exp(x^(4*k-1)).
a(n) ~ exp(sqrt(n) - n - 1/4) * n^(n - 1/4) / 2. - Vaclav Kotesovec, Oct 15 2017

A293566 E.g.f.: Product_{m>=0} exp(-x^(4*m+1)).

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
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), A293565 (k=3), this sequence (k=4).
Cf. A293507.

Programs

  • Maple
    seq(factorial(k) * coeftayl(product(exp(-x^(4*m + 1)),m = 0..k), x = 0, k),k = 0..50); # Muniru A Asiru, Oct 15 2017
  • Mathematica
    CoefficientList[Series[E^(x/(x^4 - 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^4-1))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(4*m+1)))))

Formula

E.g.f.: exp(x/(x^4 - 1)).
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

A373541 Expansion of e.g.f. exp(x/(1 + x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, 347761, 5412961, 39861361, 199488961, -5448797639, -171814122479, -2172192782759, -17416977701759, 251989368889441, 15508068901698241, 302085533813309281, 3540741711640110721, -21311122660066333079
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2024

Keywords

Crossrefs

Cf. A293507.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k-1,k)/(n-4*k)!.
a(n) == 1 mod 120.
Showing 1-7 of 7 results.

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