OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
O.g.f.: Sum_{n>=0} x^n / (1 - n^2*x).
E.g.f.: Sum_{n>=0} Integral^n exp(n^2*x) dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration.
a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021
EXAMPLE
O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 163*x^5 + 1268*x^6 +...
O.g.f.: A(x) = 1 + x/(1-x) + x^2/(1-4*x) + x^3/(1-9*x) + x^4/(1-16*x) +...
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 27*x^4/4! + 163*x^5/5! +...
where the e.g.f. is a series involving iterated integration:
E(x) = 1 + Integral exp(x) dx + Integral^2 exp(4*x) dx^2 + Integral^3 exp(9*x) dx^3 + Integral^4 exp(16*x) dx^4 +...
MATHEMATICA
Flatten[{1, Table[Sum[(n-k)^(2*k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)
PROG
(PARI) a(n)=sum(k=0, n, (n-k)^(2*k))
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From o.g.f. Sum_{n>=0} x^n/(1-n^2*x): */
{a(n)=polcoeff(sum(m=0, n, x^m/(1-m^2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From e.g.f. involving iterated integration: */
INTEGRATE(n, F)=local(G=F); for(i=1, n, G=intformal(G)); G
a(n)=my(A=1+x); A=1+sum(k=1, n, INTEGRATE(k, exp(k^2*x+x*O(x^n)))); n!*polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 28 2013
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Dec 28 2013
STATUS
approved