OFFSET
1,2
FORMULA
E.g.f.: Sum_{n>=1} (-LambertW(-y*x^n)/y)^n / (1 + LambertW(-y*x^n)) / n! at y=2.
E.g.f.: Sum_{n>=1} x^(n^2) / n! * Sum_{k>=0} (n+k)^k * y^k * x^(n*k) / k! at y=2.
...
a(n) = Sum_{d|n} (y*d)^(d-n/d) * binomial(d, n/d) * n!/d! for n>=1 at y=2.
EXAMPLE
E.g.f.: A(x) = x + 8*x^2/2! + 108*x^3/3! + 2060*x^4/4! + 50000*x^5/5! +...
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=2.
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=2.
Equivalently,
A(x) = x + 4*y*x^2/2! + 27*y^2*x^3/3! +
(256*y^3 + 12)*x^4/4! +
3125*y^4*x^5/5! +
(46656*y^5 + 1080*y)*x^6/6! +
823543*y^6*x^7/7! +
(16777216*y^7 + 161280*y^2)*x^8/8! +
(387420489*y^8 + 60480)*x^9/9! +
(10000000000*y^9 + 37800000*y^3)*x^10/10! +
285311670611*y^10*x^11/11! +
(8916100448256*y^11 + 12933043200*y^4 + 319334400*y)*x^12/12! +...
evaluated at y=2.
PROG
(PARI) a(n, y=2) = 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=2) = 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=2) = 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 sanity check): */
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 26 2015
STATUS
approved