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A265268 E.g.f.: Sum_{n>=1} LambertW(x^n)^n / (1 + LambertW(x^n)) / n!. 2
1, -4, 27, -244, 3125, -47736, 823543, -16615936, 387480969, -10037800000, 285311670611, -8903486739456, 302875106592253, -11118111848642176, 437896614702459375, -18442934324265250816, 827240261886336764177, -39349484421578544973824, 1978419655660313589123979, -104854583515276985036800000, 5842587870256483592730884421, -341431529170492630491871811584, 20880467999847912034355032910567 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..23.

FORMULA

E.g.f.: Sum_{n>=0} ((x^n - n)^n - (-n)^n) / n!.

E.g.f.: Sum_{n>=1} x^(n^2) / n! * Sum_{k>=0} (-1)^k * (n+k)^k * x^(n*k) / k!.

...

a(n) = Sum_{d|n} (-d)^(d-n/d) * binomial(d, n/d) * n!/d! for n>=1.

EXAMPLE

E.g.f.: A(x) = x - 4*x^2/2! + 27*x^3/3! - 244*x^4/4! + 3125*x^5/5! - 47736*x^6/6! + 823543*x^7/7! - 16615936*x^8/8! + 387480969*x^9/9! - 10037800000*x^10/10! +...

such that

A(x) = x + ((x^2 - 2)^2 - 2^2)/2! + ((x^3 - 3)^3 + 3^3)/3! + ((x^4 - 4)^4 - 4^4)/4! + ((x^5 - 5)^5 + 5^5)/5! + ((x^6 - 6)^6 - 6^6)/6! + ((x^7 - 7)^7 + 7^7)/7! + ((x^8 - 8)^8 - 8^8)/8! + ((x^9 - 9)^9 + 9^9)/9! +...

Also, we have the identity related to the LambertW function:

A(x) = x*[Sum_{k>=0} (-1)^k * (k+1)^k * x^k/k!] +

x^4/2!*[Sum_{k>=0} (-1)^k * (k+2)^k * x^(2*k)/k!] +

x^9/3!*[Sum_{k>=0} (-1)^k * (k+3)^k * x^(3*k)/k!] +

x^16/4!*[Sum_{k>=0} (-1)^k * (k+4)^k * x^(4*k)/k!] +

x^25/5!*[Sum_{k>=0} (-1)^k * (k+5)^k * x^(5*k)/k!] +...

PROG

(PARI) a(n) = my(A=1); A = sum(m=1, n, x^(m^2) * sum(k=0, n, (-1)^k*(k+m)^k*x^(m*k)/k! +x*O(x^n)) / m!); n!*polcoeff(A, n)

for(n=1, 30, print1(a(n), ", "))

(PARI) a(n) = my(A=1); A = sum(m=0, n, ((x^m - m +x*O(x^n))^m - (-m)^m)/m!); if(n==0, 0, n!*polcoeff(A, n))

for(n=1, 30, print1(a(n), ", "))

(PARI) a(n) = if(n<1, 0, sumdiv(n, d, (-d)^(d-n/d) * binomial(d, n/d) * n!/d! ) )

for(n=1, 30, print1(a(n), ", "))

(PARI) /* Compare these series (informal): */

LW=serreverse(x*exp(x +O(x^26)));

sum(n=1, 26, subst(LW, x, x^n)^n/n! /(1 + subst(LW, x, x^n) ) +O(x^26))

sum(n=1, 26, ((x^n - n)^n - (-n)^n)/ n! +O(x^26))

CROSSREFS

Cf. A265270, A265277.

Sequence in context: A161120 A183430 A212559 * A121063 A229619 A051863

Adjacent sequences:  A265265 A265266 A265267 * A265269 A265270 A265271

KEYWORD

sign

AUTHOR

Paul D. Hanna, Dec 26 2015

STATUS

approved

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Last modified December 9 13:50 EST 2019. Contains 329877 sequences. (Running on oeis4.)