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.

A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 481, 3841, 38221, 464353, 5368609, 82042561, 1151767981, 20242097041, 342921513793, 6705416722369, 133590317946541, 2880298682358721, 65597610230669761, 1556262483879791233, 39569880403136366029, 1030778206965403668721
Offset: 0

Views

Author

Seiichi Manyama, Jun 11 2024

Keywords

Crossrefs

Cf. A012150, A088009, A373619.
Cf. A082579, A373578.

Programs

  • Maple
    A373620 := proc(n)
    add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A373620(n),n=0..80) ; # R. J. Mathar, Jun 11 2024
  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(2*n-3*k-1, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k)/(n-2*k)!.
a(n) == 1 mod 12.
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/48 + 2^(-5/3)*n^(1/3) + 3*2^(-4/3)*n^(2/3) - n) * n^(n - 1/6). - Vaclav Kotesovec, Jun 11 2024
D-finite with recurrence a(n) -a(n-1) -3*(n-1)*(n-2)*a(n-2) -3*(n-1)*(n-2)*a(n-3) +3*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 11 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)

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)

A373619 Expansion of e.g.f. exp(x / (1 - x^2)^(3/2)).

Original entry on oeis.org

1, 1, 1, 10, 37, 316, 2341, 21736, 237385, 2611792, 35911081, 476570656, 7654975021, 121021831360, 2196593121997, 40464132512896, 817485662059921, 17159299818547456, 382733978898335185, 8982388245979044352, 219867829220866999861, 5684505550914409716736
Offset: 0

Views

Author

Seiichi Manyama, Jun 11 2024

Keywords

Crossrefs

Cf. A012150, A088009, A373620.
Cf. A373577.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n/2-2*k-1,k)/(n-2*k)!.
a(n) == 1 mod 9.
a(n) ~ 3^(1/5) * 5^(-1/2) * exp(3^(-1/5)*n^(1/5)/4 + 5*3^(-3/5)*n^(3/5)/2 - n) * n^(n - 1/5) * (1 - 1/(10*3^(4/5)*n^(1/5))). - Vaclav Kotesovec, Jun 11 2024

A373668 Expansion of e.g.f. exp(x / (1 - x^2)^3).

Original entry on oeis.org

1, 1, 1, 19, 73, 901, 7921, 88831, 1261009, 15786793, 284515201, 4359416491, 88359404761, 1671036171949, 36734936604913, 831051144091351, 19848996799904161, 516144198653004241, 13522792578340917889, 391107276466207593283, 11295497154349628317801
Offset: 0

Views

Author

Seiichi Manyama, Jun 13 2024

Keywords

Crossrefs

Cf. A012150, A088009, A373619, A373620, A373667.
Cf. A373653, A373682.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n-5*k-1,k)/(n-2*k)!.
a(n) == 1 (mod 18).

A189054 Expansion of e.g.f. exp(x/sqrt(1-4*x^2)).

Original entry on oeis.org

1, 1, 1, 13, 49, 841, 6001, 126421, 1371553, 34081489, 503678881, 14391006301, 271223253841, 8751666000793, 201326507146129, 7238365225056421, 197024810845531201, 7810072695945382561, 245787442777437613633
Offset: 0

Views

Author

Vladimir Kruchinin, Apr 16 2011

Keywords

Links

Crossrefs

Cf. A012150.

Programs

  • Mathematica
    CoefficientList[Series[Exp[x/Sqrt[1-4*x^2]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
  • Maxima
    a(n):= n!*sum((binomial((n-2)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1))/k!, k,0,n);
  • PARI
    x='x+O('x^66); /* that many terms */
egf=exp(x/sqrt(1-4*x^2)) /* = 1 +x +1/2*x^2 +13/6*x^3 +49/24*x^4 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 22 2011 */

Formula

a(n) = n! * Sum_{k=0..n} (binomial((n-2)/2, (n-k)/2) * 2^(n-k-1) * ((-1)^(n-k)+1))/k!.
a(n) ~ (2*n)^(n-1/3) / (sqrt(3)*exp(n-3/4*(2*n)^(1/3))). - Vaclav Kotesovec, Jun 02 2013

A373667 Expansion of e.g.f. exp(x / (1 - x^2)^(5/2)).

Original entry on oeis.org

1, 1, 1, 16, 61, 676, 5701, 60376, 798841, 9635536, 160878601, 2367914176, 44902245301, 807083463616, 16799688310861, 358223448539776, 8158048770370801, 199405713714155776, 4987832102850957841, 135848995301247809536, 3737769145322321702701
Offset: 0

Views

Author

Seiichi Manyama, Jun 13 2024

Keywords

Crossrefs

Cf. A012150, A088009, A373619, A373620, A373668.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(5*n/2-4*k-1, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(5*n/2-4*k-1,k)/(n-2*k)!.
a(n) == 1 (mod 15).
Showing 1-7 of 7 results.

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

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