A390845 Expansion of e.g.f. exp( x^3/6 + x^4/24 ).

1, 0, 0, 1, 1, 0, 10, 35, 35, 280, 2100, 5775, 21175, 200200, 1051050, 4029025, 30655625, 238238000, 1262661400, 7977995025, 70443998625, 511496986000, 3354718617250, 28558676020875, 254165059022875, 1961172238025000, 16283455625937500, 156856201836709375

Offset

0,7

Comments

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

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = (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(-9/32 + 5*3^(5/4)*n^(1/4)/2^(23/4) - 3^(3/2)*sqrt(n)/2^(7/2) + n^(3/4)/6^(1/4) - 3*n/4) * n^(3*n/4) * (1 + 4639*3^(3/4) / (5*2^(49/4)*n^(1/4))). - Vaclav Kotesovec, Nov 21 2025

Mathematica

nmax = 27; CoefficientList[Series[Exp[x^3/6 + x^4/24], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = (n - 1) (n - 2) a[n - 3]/2 + (n - 1) (n - 2) (n - 3) a[n - 4]/6; Table[a[n], {n, 0, 27}]

Crossrefs

Cf. A000085, A001681, A006505, A008679, A190729, A227937, A351932, A390846.

Sequence in context: A260336 A010818 A243939 * A065195 A115149 A044087

Adjacent sequences: A390842 A390843 A390844 * A390846 A390847 A390848

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2025

Status

approved