A234568 Sum_{k=0..n} (n-k)^(2*k).

1, 1, 2, 6, 27, 163, 1268, 12344, 145653, 2036149, 33192790, 622384730, 13263528351, 318121600695, 8517247764136, 252725694989612, 8258153081400857, 295515712276222953, 11523986940937975402, 487562536078882116718, 22291094729329088403299, 1097336766599161926448779

Offset: 0

Author

Paul D. Hanna, Dec 28 2013

Keywords

• nonn

Examples

			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 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A003101, A026898, A349880, A349881.

Programs

Mathematica

Flatten[{1,Table[Sum[(n-k)^(2*k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)

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

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