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.
%I A387264 #8 Aug 28 2025 19:32:23
%S A387264 1,0,0,6,96,1200,14760,196560,2983680,52315200,1041465600,22912243200,
%T A387264 545443113600,13887294220800,376188856243200,10816657377926400,
%U A387264 329526966472704000,10612556870243328000,360307460991724646400,12857257599818926694400,480829913352068087808000
%N A387264 Expansion of e.g.f. exp(x^3/(1-x)^4).
%C A387264 For n > 0, a(n) is the number of ways to seat n people on benches and select 3 people from each bench.
%C A387264 A001805 is the number of ways if only 1 bench is used.
%F A387264 From _Vaclav Kotesovec_, Aug 25 2025: (Start)
%F A387264 For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 4*k-1)/k!.
%F A387264 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).
%F A387264 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)
%e A387264 a(6)=14760 since there are 14400 ways using one bench and 360 ways with 2 benches of 3 people each.
%t A387264 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 *)
%Y A387264 Cf. A001804, A001805, A386514, A387244.
%K A387264 nonn,new
%O A387264 0,4
%A A387264 _Enrique Navarrete_, Aug 24 2025