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.

A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

Original entry on oeis.org

1, 0, 1, 6, 15, 60, 555, 3150, 17745, 158760, 1399545, 10914750, 102920895, 1104323220, 11249313075, 119330961750, 1426411411425, 17429852840400, 213417453474225, 2791671804271350, 38524272522310575, 537569719902715500, 7732658753799054075
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A047974, A361568, A361569.
Cf. A335344, A361278.

Programs

  • Mathematica
    Table[n! * Sum[Binomial[2*k,n-2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(1+x)^2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/(2^k * k!).
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)

A361569 Expansion of e.g.f. exp(x^4/24 * (1+x)^4).

Original entry on oeis.org

1, 0, 0, 0, 1, 20, 180, 840, 1715, 2520, 88200, 1940400, 29111775, 303603300, 2188286100, 12549537000, 143029511625, 3397035642000, 71419225878000, 1170096883956000, 15075357741068625, 163540869094102500, 2025016641129982500, 40912918773391665000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A047974, A361567, A361568.
Cf. A361280.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*(1+x)^4)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*binomial(4, j-4)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(4*k,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * binomial(4,k-4) * a(n-k)/(n-k)!.
a(n) = (n-1)*(n-2)*(n-3)/24 * (4*a(n-4) + 20*(n-4)*a(n-5) + 36*(n-4)*(n-5)*a(n-6) + 28*(n-4)*(n-5)*(n-6)*a(n-7) + 8*(n-4)*(n-5)*(n-6)*(n-7)*a(n-8)). -Seiichi Manyama, Jun 16 2024

A373742 Expansion of e.g.f. exp(x^3/6 * (1 + x)).

Original entry on oeis.org

1, 0, 0, 1, 4, 0, 10, 140, 560, 280, 8400, 92400, 385000, 800800, 16816800, 169569400, 784784000, 3811808000, 68803134400, 673546473600, 3693641952000, 30454440016000, 507477434464000, 5002277568288000, 33870732912016000, 386622918281600000
Offset: 0

Views

Author

Seiichi Manyama, Jun 16 2024

Keywords

Crossrefs

Cf. A361568, A372743.
Cf. A017817.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(k,n-3*k)/(6^k * k!).
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 4*(n-3)*a(n-4)).

A373743 Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).

Original entry on oeis.org

1, 0, 0, 1, 8, 20, 10, 280, 3360, 20440, 67200, 462000, 7407400, 73673600, 482081600, 3364761400, 47311264000, 657536880000, 6586994814400, 58707179731200, 740032028736000, 11832726841936000, 161121297104768000, 1857897194273120000, 23875495204536976000
Offset: 0

Views

Author

Seiichi Manyama, Jun 16 2024

Keywords

Crossrefs

Cf. A361568, A372742.
Cf. A361278, A361567.
Cf. A264622.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*k,n-3*k)/(6^k * k!).
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 8*(n-3)*a(n-4) + 5*(n-3)*(n-4)*a(n-5)).
Showing 1-4 of 4 results.

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