A393588 - OEIS

%I #36 Apr 05 2026 11:08:56

%S 1,18,183,1386,8718,48204,242434,1134108,5011791,21157758,86041713,

%T 339220566,1303018660,4895860680,18050814180,65475234360,234149987565,

%U 827019517770,2889248827195,9996416440290,34289016225546,116710858478628,394504679199318,1325168075434356

%N Expansion of 1/((1-z)*(1-2*z)*(1-3*z))^3.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 327, 341.

%H Marko Riedel, <a href="/A393588/a393588.pdf">Symbolic evaluation of the coefficients of a Stirling number generating function by partial fractions</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (18,-141,630,-1767,3222,-3815,2826,-1188,216).

%F G.f.: 1/((1-z)*(1-2*z)*(1-3*z))^3.

%F a(n) = Sum_{a+b+c=n} S2(a+3,3) * S2(b+3,3) * S2(c+3,3) where S2 is a Stirling number of the second kind.

%p b:= proc(n, i) option remember; `if`(i=0, 1, add(

%p       Stirling2(j+3, 3)*b(n-j, i-1), j=`if`(i=1, n, 0..n)))

%p     end:

%p a:= n-> b(n, 3):

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 24 2026

%t CoefficientList[Series[1/((1-x)*(1-2*x)*(1-3*x))^3, {x, 0, 23}], x] (* _Amiram Eldar_, Feb 24 2026 *)

%Y Cf. A000392, A008277, A048993, A383841, A393589, A393590.

%K nonn,easy

%O 0,2

%A _Marko Riedel_, Feb 22 2026