A232688 - OEIS

%I #10 Mar 01 2024 02:05:14

%S 1,1,6,24,160,1100,9168,84336,868544,9806688,120754080,1608251040,

%T 23036978688,353070167424,5764504370688,99875064159360,

%U 1830107743518720,35360088597841920,718450398365755392,15313427783968370688,341652563064038062080,7962756873168407869440

%N Expansion of 1/(1-x)^(1+2*x).

%C Generally, if E.g.f. = 1/(1-x)^(1+p*x) then a(n) ~ n! * n^p/p! * (1 - p^2*log(n)/n). - _Vaclav Kotesovec_, Nov 29 2013

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

%F a(n) ~ n! * n^2/2 * (1-4*log(n)/n). - _Vaclav Kotesovec_, Nov 29 2013

%e E.g.f.: A(x) = 1 + x + 6*x^2/2! + 24*x^3/3! + 160*x^4/4! + 1100*x^5/5! + 9168*x^6/6! +...

%e where

%e A(x) = 1 + (1+2*x)*x + (1+2*x)*(2+2*x)*x^2/2! + (1+2*x)*(2+2*x)*(3+2*x)*x^3/3! + (1+2*x)*(2+2*x)*(3+2*x)*(4+2*x)*x^4/4! + (1+2*x)*(2+2*x)*(3+2*x)*(4+2*x)*(5+2*x)*x^5/5! +...

%t CoefficientList[Series[1/(1-x)^(1+2*x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 29 2013 *)

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

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-2*x), n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A073479.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 27 2013