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A294343 E.g.f.: exp( Sum_{n>=1} ( Product_{k=1..n} sigma(k) ) * x^n/n! ).

%I A294343 #7 Oct 28 2017 20:57:51
%S A294343 1,1,4,22,178,1570,18808,230980,3505468,57144700,1068295600,
%T A294343 20546428360,461887088728,10502814172696,264754450444576,
%U A294343 7060121052388720,204307337026984720,6046890989734627600,195299794016884735552,6449005971683059906144,228860773033625810367520,8414329938572105578573600,325809232939269347815692160,12955190762780437329737296960,547586316827523026196832913344
%N A294343 E.g.f.: exp( Sum_{n>=1} ( Product_{k=1..n} sigma(k) ) * x^n/n! ).
%C A294343 Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).
%H A294343 Paul D. Hanna, <a href="/A294343/b294343.txt">Table of n, a(n) for n = 0..300</a>
%e A294343 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 178*x^4/4! + 1570*x^5/5! + 18808*x^6/6! + 230980*x^7/7! + 3505468*x^8/8! + 57144700*x^9/9! + 1068295600*x^10/10! + 20546428360*x^11/11! + 461887088728*x^12/12! + 10502814172696*x^13/13! + 264754450444576*x^14/14! + 7060121052388720*x^15/15! + 204307337026984720*x^16/16! +...
%e A294343 such that
%e A294343 log(A(x)) = 1*x + 1*3*x^2/2! + 1*3*4*x^3/3! + 1*3*4*7*x^4/4! + 1*3*4*7*6*x^5/5! + 1*3*4*7*6*12*x^6/6! + 1*3*4*7*6*12*8*x^7/7! + 1*3*4*7*6*12*8*15*x^8/8! + 1*3*4*7*6*12*8*15*13*x^9/9! + 1*3*4*7*6*12*8*15*13*18*x^10/10! +...+ (Product_{k=1..n} sigma(k))*x^n/n! +...
%e A294343 explicitly,
%e A294343 log(A(x)) = x + 3*x^2/2! + 12*x^3/3! + 84*x^4/4! + 504*x^5/5! + 6048*x^6/6! + 48384*x^7/7! + 725760*x^8/8! + 9434880*x^9/9! + 169827840*x^10/10! + 2037934080*x^11/11! + 57062154240*x^12/12! +...+ A066780(n)*x^n/n! +...
%o A294343 (PARI) {a(n) = n!*polcoeff( exp( sum(m=1,n+1, prod(k=1,m, sigma(k)) * x^m/m!) +x*O(x^n)),n)}
%o A294343 for(n=0,30,print1(a(n),", "))
%Y A294343 Cf. A066780.
%K A294343 nonn
%O A294343 0,3
%A A294343 _Paul D. Hanna_, Oct 28 2017