A274268 Expansion of e.g.f. (1 + x)^4*log(1 + x).

1, 7, 26, 50, 24, -24, 48, -144, 576, -2880, 17280, -120960, 967680, -8709120, 87091200, -958003200, 11496038400, -149448499200, 2092278988800, -31384184832000, 502146957312000, -8536498274304000, 153656968937472000, -2919482409811968000, 58389648196239360000

Offset: 1

Author

Peter Bala, Jun 19 2016

Keywords

• sign
• easy

Comments

First five terms [1, 7, 26, 50, 24] form row 4 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 7*x^2/2 + 26*x^3/3! + 50*x^4/4! + 24*x^5/5! - 24*x^6/6! + ...
		

Crossrefs

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

Programs

Magma

[1,7,26,50] cat [(-1)^(n-1)*24*Factorial(n-5): n in [5..25]]; // Vincenzo Librandi, Jun 20 2016

Mathematica

CoefficientList[Series[(1+t)^4 * Log[1+t], {t, 1, 20}], t]*Range[1, 20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^(n-1)*24*(n - 5)! for n >= 5.

E.g.f.: A(x) = (1 + x)^4*log(1 + x).

Series reversion(A(x)) = exp(-1/4*T(-4*x)) - 1 = x - 7*x^2/2! + 11^2*x^3/3! - 15^3*x^4/4! + 19^4*x^5/5! - ... is the e.g.f. for a signed version of A274267, 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) = 2733/2275 + 1/(24*e). - Amiram Eldar, Feb 02 2023