A387244 -id:A387244 - cp's OEIS frontend

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800

Offset: 0

Enrique Navarrete, Aug 23 2025

For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.

A001804(n) is the number of ways if only 1 line is used.

			a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.

Cf. A001804, A364524.
Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A387244.

nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

From Vaclav Kotesovec, Aug 24 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)

A387264 Expansion of e.g.f. exp(x^3/(1-x)^4).

Original entry on oeis.org

1, 0, 0, 6, 96, 1200, 14760, 196560, 2983680, 52315200, 1041465600, 22912243200, 545443113600, 13887294220800, 376188856243200, 10816657377926400, 329526966472704000, 10612556870243328000, 360307460991724646400, 12857257599818926694400, 480829913352068087808000

Offset: 0

Enrique Navarrete, Aug 24 2025

For n > 0, a(n) is the number of ways to seat n people on benches and select 3 people from each bench.

A001805 is the number of ways if only 1 bench is used.

			a(6)=14760 since there are 14400 ways using one bench and 360 ways with 2 benches of 3 people each.

Cf. A001804, A001805, A386514, A387244.

nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 4*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 25 2025 *)

From Vaclav Kotesovec, Aug 25 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 4*k-1)/k!.
a(n) = 5*(n-1)*a(n-1) - 10*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(10*n-27)*a(n-3) - (n-3)*(n-2)*(n-1)*(5*n-21)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(-27/1280 + 13*2^(-22/5)*n^(1/5)/25 + 13*2^(-19/5)*n^(2/5)/15 - 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) * n^(n-1/10). (End)

cp's OEIS Frontend

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula