Factoriangular numbers (denoted by Ftn) are formed by adding corresponding factorials and triangular numbers, that is Ftn = n! + Tn, where Tn is the nth triangular number. The closed form of the exponential generating function of the sequence of factoriangular numbers can be easily derived from the exponential generating functions of n! and of Tn (see the link below). How about the closed form of the ordinary generating function of such sequence?
Article Recurrence Relations and Generating Functions of the Sequenc...
Dear Romer,
There are many problems with the formal series y(t)=\sum_{n\ge 1} n! t^n.
First, the radius of convergence is equal to 0. Second, one can write down formal linear differential equation on y(t), but one gets singularity at t=0 and non-elementary formal solution. Will it help?
Best regards, Yuri
In principle there is a procedure to go from egf to ogf. Check the literature in analysis of algorithms. Knuth, Flajolet etc..
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