A390846 Expansion of e.g.f. exp( x^2/2 + x^3/6 + x^4/24 ).

1, 0, 1, 1, 4, 10, 40, 140, 630, 2800, 14070, 73150, 412720, 2422420, 15095080, 98098000, 668567900, 4737532800, 34971636700, 267614447100, 2122421501200, 17397687307000, 147282162227200, 1285154736065200, 11547913098521800, 106706509549640000, 1012993221405165000

Offset

0,5

Comments

Number of set partitions of [n] into blocks of size 2, 3 or 4.

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = (n-1) * a(n-2) + (n-1) * (n-2) * a(n-3) / 2 + (n-1) * (n-2) * (n-3) * a(n-4) / 6.

a(n) ~ 2^(-1 - n/4) * 3^(-n/4) * exp(3/32 - 11*3^(5/4)*n^(1/4)/2^(23/4) + 5*sqrt(3*n)/2^(7/2) + n^(3/4)/6^(1/4) - 3*n/4) * n^(3*n/4) * (1 + 6079*3^(3/4) / (5*2^(49/4)*n^(1/4))). - Vaclav Kotesovec, Nov 21 2025

Mathematica

nmax = 26; CoefficientList[Series[Exp[x^2/2 + x^3/6 + x^4/24], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[1] = 0; a[2] = a[3] = 1; a[n_] := a[n] = (n - 1) a[n - 2] + (n - 1) (n - 2) a[n - 3]/2 + (n - 1) (n - 2) (n - 3) a[n - 4]/6; Table[a[n], {n, 0, 26}]

Crossrefs

Cf. A000296, A001680, A001681, A005044, A190452, A190729, A227937, A390845.

Sequence in context: A149211 A149212 A223160 * A263551 A007173 A222367

Adjacent sequences: A390843 A390844 A390845 * A390847 A390848 A390849

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2025

Status

approved