A394656 a(n) = binomial(n,3) + 4*binomial(n,4) + 10*binomial(n,5) + 20*binomial(n,6).

0, 0, 0, 1, 8, 40, 160, 525, 1456, 3528, 7680, 15345, 28600, 50336, 84448, 136045, 211680, 319600, 470016, 675393, 950760, 1314040, 1786400, 2392621, 3161488, 4126200, 5324800, 6800625, 8602776, 10786608, 13414240, 16555085, 20286400, 24693856, 29872128, 35925505, 42968520

Offset

0,5

Comments

a(n) is the number of ordered set partitions of an n-set into 2 sets such that the first set has either 3, 2, 1 or no elements, the second set has no restrictions, and three elements are selected from the second set.

Note that the coefficients 1,4,10,20 in formula for a(n) are triangular pyramidal numbers (in accordance with the selection of three elements from the second set).

a(n) is also the number of ternary strings of length n with exactly three 1's, at most three 0's and any number of 2's.

Links

Table of n, a(n) for n=0..36.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = n*(n-1)*(n-2)^2*(n^2-7*n+18)/36.

E.g.f.: (1 + x + x^2/2 + x^3/6)*x^3/6*exp(x).

G.f.: x^3*(1 + x + 5*x^2 + 13*x^3)/(1 - x)^7. - Stefano Spezia, May 03 2026

Example

The 525 set partitions for n=7 are the following (where the 3 elements selected from the second set are in parentheses):
  { }, {(1),(2),(3),4,5,6,7} (35 of these);
  {1}, {(2),(3),(4),5,6,7}   (140 of these);
  {1,2}, {(3),(4),(5),6,7}   (210 of these);
  {1,2,3}, {(4),(5),(6),7}   (140 of these).

Mathematica

a[n_]:=n*(n-1)*(n-2)^2*(n^2-7*n+18)/36; Array[a, 37, 0] (* Stefano Spezia, May 03 2026 *)

Crossrefs

Cf. A006000, A361099, A361474.

Sequence in context: A125198 A128639 A341365 * A004405 A284286 A001789

Adjacent sequences: A394653 A394654 A394655 * A394657 A394658 A394659

Keyword

nonn,easy

Author

Enrique Navarrete, Apr 29 2026

Status

approved