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

1, 1, 4, 22, 178, 1570, 18808, 230980, 3505468, 57144700, 1068295600, 20546428360, 461887088728, 10502814172696, 264754450444576, 7060121052388720, 204307337026984720, 6046890989734627600, 195299794016884735552, 6449005971683059906144, 228860773033625810367520, 8414329938572105578573600, 325809232939269347815692160, 12955190762780437329737296960, 547586316827523026196832913344

Offset: 0

Author

Paul D. Hanna, Oct 28 2017

Keywords

• nonn

Comments

Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).

Examples

			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! +...
such that
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! +...
explicitly,
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! +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A066780.

Programs

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)}
for(n=0,30,print1(a(n),", "))