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Expansion of e.g.f. (1 + x)^4*log(1 + x).
3

%I #20 Feb 02 2023 06:24:29

%S 1,7,26,50,24,-24,48,-144,576,-2880,17280,-120960,967680,-8709120,

%T 87091200,-958003200,11496038400,-149448499200,2092278988800,

%U -31384184832000,502146957312000,-8536498274304000,153656968937472000,-2919482409811968000,58389648196239360000

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

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

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

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

%F 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.

%F Sum_{n>=1} 1/a(n) = 2733/2275 + 1/(24*e). - _Amiram Eldar_, Feb 02 2023

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

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

%o (Magma) [1,7,26,50] cat [(-1)^(n-1)*24*Factorial(n-5): n in [5..25]]; // _Vincenzo Librandi_, Jun 20 2016

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

%K sign,easy

%O 1,2

%A _Peter Bala_, Jun 19 2016