A265270 E.g.f.: Sum_{n>=0} (n*y + x^n)^n / n! - Sum_{n>=0} n^n*y^n / n! at y=1.

1, 4, 27, 268, 3125, 47736, 823543, 16938496, 387480969, 10037800000, 285311670611, 8929352825856, 302875106592253, 11118111848642176, 437896614702459375, 18450553823153852416, 827240261886336764177, 39349484421578544973824, 1978419655660313589123979, 104860617498432185036800000, 5842587870256483592730884421, 341431529170492630491871811584, 20880467999847912034355032910567

Offset

1,2

Links

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

Formula

E.g.f.: Sum_{n>=1} (-LambertW(-y*x^n)/y)^n / (1 + LambertW(-y*x^n)) / n! at y=1.

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

...

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

Example

E.g.f.: A(x) = x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3125*x^5/5! + 47736*x^6/6! + 823543*x^7/7! + 16938496*x^8/8! + 387480969*x^9/9! + 10037800000*x^10/10! + ...
such that
A(x) = [(y + x) + (2*y + x^2)^2/2! + (3*y + x^3)^3/3! + (4*y + x^4)^4/4! + (5*y + x^5)^5/5! + (6*y + x^6)^6/6! + (7*y + x^7)^7/7! + ...]
  - [y + 2^2*y^2/2! + 3^3*y^3/3! + 4^4*y^4/4! + 5^5*y^5/5! + 6^6*y^6/6! + ...]
evaluated at y=1.
Also, we have the identity related to the LambertW function:
A(x) = x*[Sum_{k>=0} (k+1)^k * y^k * x^k/k!] +
  x^4/2!*[Sum_{k>=0} (k+2)^k * y^k * x^(2*k)/k!] +
  x^9/3!*[Sum_{k>=0} (k+3)^k * y^k * x^(3*k)/k!] +
  x^16/4!*[Sum_{k>=0} (k+4)^k * y^k * x^(4*k)/k!] +
  x^25/5!*[Sum_{k>=0} (k+5)^k * y^k * x^(5*k)/k!] + ...
evaluated at y=1.

Mathematica

a[n_] := DivisorSum[n, #^(#-n/#) * Binomial[#, n/#] * n!/#! &]; Array[a, 25] (* Amiram Eldar, Aug 24 2023 *)

Prog

(PARI) a(n, y=1) = my(A=1); A = sum(m=1, n, x^(m^2) * sum(k=0, n, (k+m)^k*y^k*x^(m*k)/k! +x*O(x^n)) / m!); n!*polcoeff(A, n)
for(n=1, 30, print1(a(n), ", "))
(PARI) a(n, y=1) = my(A=1); A = sum(m=0, n, ((m*y + x^m +x*O(x^n))^m - m^m*y^m)/m!); if(n==0, 0, n!*polcoeff(A, n))
for(n=1, 30, print1(a(n), ", "))
(PARI) a(n, y=1) = if(n<1, 0, sumdiv(n, d, (d*y)^(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, ((n*y + x^n)^n - n^n*y^n)/ n! +O(x^26))
sum(n=1, 26, (-subst(LW, x, -x^n*y)/y)^n/n! /(1 + subst(LW, x, -x^n*y) ) +O(x^26))

Crossrefs

Cf. A265277 (y=2), A265268 (y=-1), A259209, A259223, A265943, A265269.

Sequence in context: A121353 A331316 A353233 * A161633 A052871 A104653

Adjacent sequences: A265267 A265268 A265269 * A265271 A265272 A265273

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2015

Status

approved