A274266 Expansion of e.g.f. (1 + x)^3*log(1 + x).

1, 5, 11, 6, -6, 12, -36, 144, -720, 4320, -30240, 241920, -2177280, 21772800, -239500800, 2874009600, -37362124800, 523069747200, -7846046208000, 125536739328000, -2134124568576000, 38414242234368000, -729870602452992000, 14597412049059840000

Offset: 1

Author

Peter Bala, Jun 19 2016

Keywords

• sign
• easy

Comments

First four terms [1, 5, 11, 6] form row 3 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 5*x^2/2 + 11*x^3/3! + 6*x^4/4! - 6*x^5/5! + ....
		

Crossrefs

Cf. A000169, A105954, A274265, A274268, A274270.

Programs

Magma

[1,5,11] cat [(-1)^n*6*Factorial(n-4): n in [4..25]]; // Vincenzo Librandi, Jun 20 2016

Mathematica

CoefficientList[Series[(1+t)^3 * Log[1+t], {t, 1, 20}], t]*Range[1, 20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^n*6*(n - 4)! for n >= 4.

E.g.f.: A(x) = (1 + x)^3*log(1 + x).

Series reversion(A(x)) = exp(-1/3*T(-3*x)) - 1 = x - 5*x^2/2! + 8^2*x^3/3! - 11^3*x^4/4! + 14^4*x^5/5! - ... is the e.g.f. for a signed version of A274265, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.

Sum_{n>=1} 1/a(n) = 71/55 + 1/(6*e). - Amiram Eldar, Feb 02 2023