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 1-4 of 4 results.

A361597 Expansion of e.g.f. exp( x^3/(6 * (1-x)^3) ) / (1-x).

Original entry on oeis.org

1, 1, 2, 7, 40, 320, 3130, 34930, 432320, 5866840, 86816800, 1395455600, 24270908200, 454897042600, 9146979842000, 196443726879400, 4486709145318400, 108548344109004800, 2771885136281060800, 74475606190225240000, 2099591224223100608000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A002720, A361596.
Cf. A361573.

Programs

  • Mathematica
    Table[n! * Sum[Binomial[n,3*k]/(6^k * k!), {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))/(1-x)))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/(6^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
Recurrence: 2*a(n) = 2*(4*n - 3)*a(n-1) - 6*(n-1)*(2*n - 3)*a(n-2) + (n-2)*(n-1)*(8*n - 17)*a(n-3) - 2*(n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(-7/8) * exp(-1/24 + 5*2^(-15/4)*n^(1/4)/3 - sqrt(n/2)/2 + 2^(7/4)*n^(3/4)/3 - n) * n^(n + 1/8) * (1 + (2637/10240)*2^(3/4)/n^(1/4)). (End)

A373770 Expansion of e.g.f. exp(x^2 / (2 * (1 - x))) / (1 - x).

Original entry on oeis.org

1, 1, 3, 12, 63, 405, 3075, 26880, 265545, 2922885, 35447895, 469396620, 6736095135, 104102463465, 1723322736135, 30416726340000, 570089983287825, 11306156398562025, 236514323713142475, 5204122351983254700, 120139520273298100575, 2903216115946088267325
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130905, A361596, A373771.
Cf. A185369.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k,n-2*k)/(2^k * k!).
From Vaclav Kotesovec, Jun 18 2024: (Start)
Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*a(n-3).
a(n) ~ 2^(-1/4) * exp(-3/4 + sqrt(2*n) - n) * n^(n + 1/4) * (1 + 7/(6*sqrt(2*n))). (End)

A373771 Expansion of e.g.f. exp(x^2 / (2 * (1 - x)^3)) / (1 - x).

Original entry on oeis.org

1, 1, 3, 18, 147, 1425, 15855, 200130, 2838465, 44767485, 777046095, 14705245170, 301014595035, 6621102973485, 155640761791515, 3891902825660850, 103115436832433025, 2884715829245475225, 84950805438277854075, 2626194012669689512050
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130905, A361596, A373770.
Cf. A361597, A361599.
Cf. A335345.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k,n-2*k)/(2^k * k!).

A373773 Expansion of e.g.f. exp(x^3 / (6 * (1 - x)^2)) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 36, 240, 1930, 17990, 189840, 2233000, 28949200, 410009600, 6297999400, 104275571400, 1851050401200, 35065930299400, 705993054166400, 15051593241484800, 338705933426660800, 8021585392026606400, 199416162740963168000, 5191567315003621552000
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130906, A361597, A373772.
Cf. A361596, A361598.
Cf. A373757.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-k, n-3*k)/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k,n-3*k)/(6^k * k!).
From Vaclav Kotesovec, Jun 18 2024: (Start)
Recurrence: 6*a(n) = 6*(3*n-2)*a(n-1) - 6*(n-1)*(3*n-4)*a(n-2) + 3*(n-2)*(n-1)*(2*n-3)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/3) * exp(19/72 - 3^(-2/3)*n^(1/3) + 3^(2/3)*n^(2/3)/2 - n) * n^(n + 1/6). (End)
Showing 1-4 of 4 results.

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