There are many relationships or connections between factorials and triangular numbers that can be found in the literature, both old and new. In a recent study on factoriangular numbers (sums of corresponding factorials and triangular numbers), we unintentionally found that n! = T(n-1)!+n - T(n-1)! - Tn where Tk is the kth triangular number. Curious on this, can you help us find older literature that also mentioned the stated relationship between factorials and triangular numbers?
Thank you Prof. Tsambani. I read the work of Hoggatt Jr. and Bicknell on Triangular Numbers (the document you attach) but did not find the relation between factorials and triangular numbers that I am looking for. Please feel free to suggest other documents. Thank you again.
Because
T(n-1)!+n=[(n-1)!+n][(n-1)!+n+1]/2
T(n-1)!=(n-1)![(n-1)!+1]/2
Tn=n(n+1)/2.
Therefore
T(n-1)!+n-T(n-1)!-Tn=n(n-1)!=n!
Prof. Nianrong Feng: Perhaps, you may be interested to read the following article, where the above-mentioned relation between factorials and triangular numbers was included (see Corollary 3.1 on p.99):
Article On the Sum of Corresponding Factorials and Triangular Number...
Factorials are not relevant here. For any numbers m and n one has the identity
mn = T(m+n) - T(m) - T(n), where T(x) = x(x+1)/2
Of course, one can use m=n! as well.
Some more similar relations:
mn = T(m) + T(n-1) - T(m-n)
mn = T(m-1) + T(n) - T(n-m)
This means, in particular, that one can obtain the product of any two positive integers m and n each not larger than N by precomputing the numbers T(0), T(1), ... T(N) and then doing only few additions / subtractions.
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