A274268 -id:A274268 - cp's OEIS frontend

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

Peter Bala, Jun 19 2016

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

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

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

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

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

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

A274267 a(n) = (4*n - 1)^(n-1).

Original entry on oeis.org

1, 7, 121, 3375, 130321, 6436343, 387420489, 27512614111, 2251875390625, 208728361158759, 21611482313284249, 2472159215084012303, 309629344375621415601, 42141982597572021484375, 6193386212891813387462761, 977480813971145474830595007, 164890958756244164895763202881

Offset: 1

Peter Bala, Jun 19 2016

Compare with A052774.

G. C. Greubel, Table of n, a(n) for n = 1..321

Cf. A000169, A052774, A274265, A274266, A274268, A274269.

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

A274267 := n -> (4*n - 1)^(n-1):
seq(A274267(n), n = 1..20);

Table[(4*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)

for(n=1,30, print1((4*n-1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

E.g.f. A(x) = 1 - exp(-1/4*T(4*x)) = x + 7*x^2/2! + 11^2*x^3/3! + 15^3*x^4/4! + 19^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^4*log(1/(1 - x)) ). See A274268.
1 - A(x) = exp(-x/(1 - A(x))^4) = exp(-x/(exp(-4*x/(exp(-4*x/ ...))))).
1 - A(-x*exp(4*x)) = exp(x) = 1/(1 - A(x*exp(-4*x))).
1/(1 - A(x)) = Sum_{n >= 0} (4*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052774.

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

Original entry on oeis.org

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

Peter Bala, Jun 19 2016

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

			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! + ....

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

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

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

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

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

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A274267 a(n) = (4*n - 1)^(n-1).

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

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