A058645 a(n) = 2^(n-3)*n^2*(n+3).

0, 1, 10, 54, 224, 800, 2592, 7840, 22528, 62208, 166400, 433664, 1105920, 2768896, 6823936, 16588800, 39845888, 94699520, 222953472, 520486912, 1205862400, 2774532096, 6343884800, 14422114304, 32614907904, 73400320000

Offset: 0

Author

Yong Kong (ykong(AT)curagen.com), Dec 26 2000

Keywords

• nonn
• easy

Comments

a(n) is the number of ways to select a subset of {1,2,...n} and then use the subset as an alphabet to form ordered triples.

References

• A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Crossrefs

First differences are in A084903.

Cf. A000578, A001793.

Programs

Mathematica

CoefficientList[Series[(x+3x^2+x^3) Exp[x]^2,{x,0,20}],x] * Table[n!,{n,0,20}]

PARI

a(n)=2^(n-3)*n^2*(n+3)

Formula

a(n) = Sum_{k=0..n} k^3 * binomial(n, k): binomial transform of A000578.

G.f.: x*(1+2*x-2*x^2)/(1-2*x)^4. E.g.f.: x*(1+3*x+x^2)*e^(2*x).

A001793(n)*(n+3) = -a(-3-n)*2^(2*n+3) for all n in Z. - Michael Somos, Apr 19 2019