A274270 -id:A274270 - 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

A274269 a(n) = (5*n - 1)^(n-1).

Original entry on oeis.org

1, 9, 196, 6859, 331776, 20511149, 1544804416, 137231006679, 14048223625216, 1628413597910449, 210832519264920576, 30155888444737842659, 4722366482869645213696, 803596764671634487466709, 147653612273582215982104576, 29134419507545592909032289199

Offset: 1

Peter Bala, Jun 19 2016

Compare with A052782.

Cf. A000169, A052782, A274265, A274267, A274270.

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

A274269 := n -> (5*n - 1)^(n-1):
seq(A274269(n), n = 1..20);

Table[(5*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)

E.g.f. A(x) = 1 - exp(-1/5*T(5*x)) = x + 9*x^2/2! + 14^2*x^3/3! + 19^3*x^4/4! + 24^4*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)^5*log(1/(1 - x)) ). See A274270.
1 - A(x) = exp(-x/(1 - A(x))^5) = exp(-x/(exp(-5*x/(exp(-5*x/ ...))))).
1 - A(-x*exp(5*x)) = exp(x) = 1/(1 - A(x*exp(-5*x))).
1/(1 - A(x)) = Sum_{n >= 0} (5*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052782.

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

Original entry on oeis.org

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

Peter Bala, Jun 19 2016

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

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

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

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

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

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

cp's OEIS Frontend

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

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A274269 a(n) = (5*n - 1)^(n-1).

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

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