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A274270
Expansion of e.g.f. (1 + x)^5*log(1 + x).
4
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
1,2
COMMENTS
The first six terms [1, 9, 47, 154, 274, 120] form row 5 of A105954 read as a triangular array.
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
EXAMPLE
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! + ....
MATHEMATICA
CoefficientList[Series[(1+t)^5 * Log[1+t], {t, 1, 20}], t]*Range[1, 20]! (* G. C. Greubel, Jun 19 2016 *)
PROG
(Magma) [1, 9, 47, 154, 274] cat [(-1)^n*120*Factorial(n - 6): n in [6..25]]; // Vincenzo Librandi, Jun 20 2016
KEYWORD
sign,easy
AUTHOR
Peter Bala, Jun 19 2016
STATUS
approved