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a(n) = n!*(e*Gamma(n,1)*(n-1)+1).
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%I #23 Sep 29 2015 09:45:33

%S 1,6,66,1176,31320,1174320,59184720,3866728320,318176449920,

%T 32215365100800,3937433507884800,571715345296972800,

%U 97295556944518732800,19183440644220345292800,4338408884154346729728000,1115590855925401950302208000,323670093665823262135185408000,105233239203100787701553799168000

%N a(n) = n!*(e*Gamma(n,1)*(n-1)+1).

%F a(n) is the sum of infinite series of the hypergeometric functions of type 2F0, in Maple notation: a(n)=sum(k*(n+k-1)!*hypergeom([k+1,k+1],[],-1),k=1..infinity),n=1,2,... .

%F a(n) = n!*(A000522(n-1)*(n-1)+1). - _Jean-François Alcover_, Sep 26 2015

%p seq(simplify(n!*(n*exp(1)*GAMMA(n,1)-exp(1)*GAMMA(n,1)+1)),n=1..18);

%p # GAMMA(n,1) is the incomplete gamma function.

%Y Cf. A000522.

%K nonn

%O 1,2

%A _Karol A. Penson_ and Katarzyna Gorska, Sep 25 2015