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A294346 E.g.f.: exp( Sum_{n>=1} sigma(n!) * x^n/n! ).

%I A294346 #11 Oct 29 2017 06:37:58
%S A294346 1,1,4,22,154,1306,12874,145954,1843660,25840684,397040064,6637371896,
%T A294346 119517187984,2310108276048,47619441310520,1042743337601320,
%U A294346 24164137431011184,590726322945970352,15184954152657360064,409428979786326488096,11550423660014156192096,340219279585618435264480,10442779307230643663779424,333425628200639984852617568,11055838405832227887079632832
%N A294346 E.g.f.: exp( Sum_{n>=1} sigma(n!) * x^n/n! ).
%C A294346 Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).
%H A294346 Paul D. Hanna, <a href="/A294346/b294346.txt">Table of n, a(n) for n = 0..300</a>
%e A294346 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 154*x^4/4! + 1306*x^5/5! + 12874*x^6/6! + 145954*x^7/7! + 1843660*x^8/8! + 25840684*x^9/9! + 397040064*x^10/10! + 6637371896*x^11/11! + 119517187984*x^12/12! +...
%e A294346 such that
%e A294346 log(A(x)) = x + sigma(2!)*x^2/2! + sigma(3!)*x^3/3! + sigma(4!)*x^4/4! + sigma(5!)*x^5/5! + sigma(6!)*x^6/6! +...+ sigma(n!)*x^n/n! +...
%e A294346 Explicitly,
%e A294346 log(A(x)) = x + 3*x^2/2! + 12*x^3/3! + 60*x^4/4! + 360*x^5/5! + 2418*x^6/6! + 19344*x^7/7! + 159120*x^8/8! + 1481040*x^9/9! + 15334088*x^10/10! + 184009056*x^11/11! + 2217441408*x^12/12! +...+ A062569(n)*x^n/n! +...
%e A294346 PRODUCT.
%e A294346 A(x) = 1 / ((1-x) * (1-x^2)^(2/2!) * (1-x^3)^(10/3!) * (1-x^4)^(42/4!) * (1-x^5)^(336/5!) * (1-x^6)^(1458/6!) * (1-x^7)^(18624/7!) * (1-x^8)^(108720/8!) * (1-x^9)^(1239120/9!) * (1-x^10)^(9165128/10!) * (1-x^11)^(180380256/11!) * (1-x^12)^(1133700288/12!) *...).
%o A294346 (PARI) {a(n) = n!*polcoeff( exp( sum(m=1, n+1, sigma(m!) * x^m/m!) +x*O(x^n)), n)}
%o A294346 for(n=0, 30, print1(a(n), ", "))
%Y A294346 Cf. A062569.
%K A294346 nonn
%O A294346 0,3
%A A294346 _Paul D. Hanna_, Oct 29 2017