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.

A361596 Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

Original entry on oeis.org

1, 1, 3, 15, 99, 795, 7485, 80745, 981225, 13253625, 196834995, 3185662095, 55770765435, 1049572599075, 21120725230605, 452384160453225, 10272547048388625, 246434674107647025, 6226347228582355875, 165224032352989584975, 4593512876411509125075
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A002720, A361597.
Cf. A335344.

Programs

  • Mathematica
    Table[n! * Sum[Binomial[n,2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*(1-x)^2))/(1-x)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,2*k)/(2^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 2)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 3^(-1/2) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) * n^(n + 1/6) * (1 + 49/(108*n^(1/3)) + 3293/(116640*n^(2/3))). (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!).

A373772 Expansion of e.g.f. exp(x^3 / (6 * (1 - x))) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 32, 180, 1210, 9450, 84000, 836920, 9234400, 111742400, 1471023400, 20925905000, 319830310800, 5226116295400, 90906373958400, 1676967192700800, 32697692264036800, 671856896755844800, 14509136903381120000, 328520930667097168000
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130906, A361597, A373773.
Cf. A000930, A361533.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-2*k, n-3*k)/(6^k*k!));

Formula

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

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

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