A393588 Expansion of 1/((1-z)*(1-2*z)*(1-3*z))^3.

1, 18, 183, 1386, 8718, 48204, 242434, 1134108, 5011791, 21157758, 86041713, 339220566, 1303018660, 4895860680, 18050814180, 65475234360, 234149987565, 827019517770, 2889248827195, 9996416440290, 34289016225546, 116710858478628, 394504679199318, 1325168075434356

Offset

0,2

References

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

Links

Table of n, a(n) for n=0..23.

Marko Riedel, Symbolic evaluation of the coefficients of a Stirling number generating function by partial fractions.

Index entries for linear recurrences with constant coefficients, signature (18,-141,630,-1767,3222,-3815,2826,-1188,216).

Formula

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

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.

Maple

b:= proc(n, i) option remember; `if`(i=0, 1, add(
      Stirling2(j+3, 3)*b(n-j, i-1), j=`if`(i=1, n, 0..n)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 24 2026

Mathematica

CoefficientList[Series[1/((1-x)*(1-2*x)*(1-3*x))^3, {x, 0, 23}], x] (* Amiram Eldar, Feb 24 2026 *)

Crossrefs

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

Sequence in context: A155669 A160954 A160955 * A026091 A178260 A296778

Adjacent sequences: A393585 A393586 A393587 * A393589 A393590 A393591

Keyword

nonn,easy

Author

Marko Riedel, Feb 22 2026

Status

approved