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Sum_{k=0..n} (n-k)^(2*k).
7

%I #21 Nov 14 2024 17:31:37

%S 1,1,2,6,27,163,1268,12344,145653,2036149,33192790,622384730,

%T 13263528351,318121600695,8517247764136,252725694989612,

%U 8258153081400857,295515712276222953,11523986940937975402,487562536078882116718,22291094729329088403299,1097336766599161926448779

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

%H Vincenzo Librandi, <a href="/A234568/b234568.txt">Table of n, a(n) for n = 0..200</a>

%F O.g.f.: Sum_{n>=0} x^n / (1 - n^2*x).

%F 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.

%F 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

%e O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 163*x^5 + 1268*x^6 +...

%e 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 E.g.f.: E(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 27*x^4/4! + 163*x^5/5! +...

%e where the e.g.f. is a series involving iterated integration:

%e 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 +...

%t Flatten[{1,Table[Sum[(n-k)^(2*k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 23 2014 *)

%o (PARI) a(n)=sum(k=0, n, (n-k)^(2*k))

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* From o.g.f. Sum_{n>=0} x^n/(1-n^2*x): */

%o {a(n)=polcoeff(sum(m=0, n, x^m/(1-m^2*x+x*O(x^n))), n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* From e.g.f. involving iterated integration: */

%o INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G

%o 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)

%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 28 2013

%Y Cf. A003101, A026898, A349880, A349881.

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Dec 28 2013