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

1, 1, 4, 22, 154, 1306, 12874, 145954, 1843660, 25840684, 397040064, 6637371896, 119517187984, 2310108276048, 47619441310520, 1042743337601320, 24164137431011184, 590726322945970352, 15184954152657360064, 409428979786326488096, 11550423660014156192096, 340219279585618435264480, 10442779307230643663779424, 333425628200639984852617568, 11055838405832227887079632832

Offset

0,3

Comments

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

Links

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

Example

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! +...
such that
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! +...
Explicitly,
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! +...
PRODUCT.
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!) *...).

Prog

(PARI) {a(n) = n!*polcoeff( exp( sum(m=1, n+1, sigma(m!) * x^m/m!) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A062569.

Sequence in context: A062817 A196275 A000307 * A049376 A083410 A295553

Adjacent sequences: A294343 A294344 A294345 * A294347 A294348 A294349

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2017

Status

approved