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A292807 E.g.f.: Sum_{n=-oo..+oo} x^n * exp(n*x) * (exp(n*x) - x^n)^n. 1
1, 7, 37, 807, 13441, 413243, 13468813, 571503103, 28308826657, 1666118229819, 113262531063661, 8830681086125231, 780324383486361793, 77494753844884990123, 8581533546227249944141, 1052503537117606772557695, 142116165804218004169556929, 21014208913195247508525503483, 3386598011981006005953444008269, 592290509726367692126040767254639 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare e.g.f. to: Sum_{n=-oo..+oo} x^n * exp(n*x) * (1 - x^n*exp(n*x))^n = 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..100

FORMULA

E.g.f.: Sum_{n=-oo..+oo} (-1)^n * x^(n^2-n) * exp((n^2-n)*x) / (exp(n*x) - x^n)^n.

EXAMPLE

E.g.f: A(x) = x + 7*x^2/2! + 37*x^3/3! + 807*x^4/4! + 13441*x^5/5! + 413243*x^6/6! + 13468813*x^7/7! + 571503103*x^8/8! + 28308826657*x^9/9! + 1666118229819*x^10/10! +...

Let E = exp(x), then A(x) = P(x) + Q(x) where

P(x) = 1 + (x*E)*(E - x) + (x*E)^2*(E^2 - x^2)^2 + (x*E)^3*(E^3 - x^3)^3 + (x*E)^4*(E^4 - x^4)^4 + (x*E)^5*(E^5 - x^5)^5 +...+ (x*E)^n*(E^n - x^n)^n +...

Q(x) = -1/(E - x) + (x*E)^2/(E^2 - x^2)^2 - (x*E)^6/(E^3 - x^3)^3 + (x*E)^12/(E^4 - x^4)^4 - (x*E)^20/(E^5 - x^5)^5 +...+ (-1)^n*(x*E)^(n^2-n)/(E^n - x^n)^n +...

Explicitly,

P(x) = 1 + x + 4*x^2/2! + 48*x^3/3! + 716*x^4/4! + 14580*x^5/5! + 399762*x^6/6! + 13652758*x^7/7! + 568482056*x^8/8! + 28365307128*x^9/9! + 1664953425350*x^10/10! +...

Q(x) = -1 + 3*x^2/2! - 11*x^3/3! + 91*x^4/4! - 1139*x^5/5! + 13481*x^6/6! - 183945*x^7/7! + 3021047*x^8/8! - 56480471*x^9/9! + 1164804469*x^10/10! +...

PROG

(PARI) {a(n) = my(A, P, Q, E=exp(x + x*O(x^n)));

P = sum(m=0, n, (x*E)^m*(E^m - x^m)^m);

Q = sum(m=1, n, (-1)^m*(x*E)^(m^2-m)/(E^m - x^m)^m);

A = P + Q; n!*polcoeff(A, n)}

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

CROSSREFS

Cf. A292088.

Sequence in context: A082687 A117731 A155010 * A210620 A250843 A078303

Adjacent sequences:  A292804 A292805 A292806 * A292808 A292809 A292810

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 04 2017

STATUS

approved

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Last modified April 18 15:04 EDT 2021. Contains 343089 sequences. (Running on oeis4.)