A274270 Expansion of e.g.f. (1 + x)^5*log(1 + x).

1, 9, 47, 154, 274, 120, -120, 240, -720, 2880, -14400, 86400, -604800, 4838400, -43545600, 435456000, -4790016000, 57480192000, -747242496000, 10461394944000, -156920924160000, 2510734786560000, -42682491371520000, 768284844687360000, -14597412049059840000

Offset: 1

Author

Peter Bala, Jun 19 2016

Keywords

• sign
• easy

Comments

The first six terms [1, 9, 47, 154, 274, 120] form row 5 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 9*x^2/2 + 47*x^3/3! + 154*x^4/4! + 274*x^5/5! + 120*x^6/6! - 120*x^7/7! + ....
		

Crossrefs

Cf. A000169, A105954, A274265, A274266, A274267, A274268, A274269, A274270.

Programs

Magma

[1,9,47,154,274] cat [(-1)^n*120*Factorial(n - 6): n in [6..25]]; // Vincenzo Librandi, Jun 20 2016

Mathematica

CoefficientList[Series[(1+t)^5 * Log[1+t], {t,1,20}],t]*Range[1,20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^n*120*(n - 6)! for n >= 6.

E.g.f.: A(x) = (1 + x)^5*log(1 + x).

Series reversion(A(x)) = exp(-1/5*T(-5*x)) - 1 = x - 9*x^2/2! + 14^2*x^3/3! - 19^3*x^4/4! + 24^4*x^5/5! - ... is the e.g.f. for a signed version of A274269, 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) = 5098232/4462227 + 1/(120*e). - Amiram Eldar, Feb 02 2023