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

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

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..20.

Formula

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)

Example

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

Mathematica

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 *)

Crossrefs

Cf. A001804, A001805, A386514, A387244.

Sequence in context: A202078 A227262 A001805 * A385903 A139743 A361737

Adjacent sequences: A387261 A387262 A387263 * A387265 A387266 A387267

Keyword

nonn

Author

Enrique Navarrete, Aug 24 2025

Status

approved