A294352 Product of first n terms of the binomial transform of the factorial.
1, 2, 10, 160, 10400, 3390400, 6635012800, 90899675360000, 9962695319131360000, 9827302289744364817600000, 96937502343569678741652977600000, 10518214548789290471667075399621491200000, 13695360582395151673134516587047571322777664000000
Offset: 0
Keywords
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..43
- M. Bahrami-Taghanaki, A. R. Moghaddamfar, Nima Salehy, and Navid Salehy, Some Identities Involving Stirling Numbers Arising from Matrix Decompositions, J. Int. Seq. (2024) Vol. 24, No. 5, Art. No. 24.5.3. See p. 9.
Programs
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Mathematica
Table[Product[Sum[Binomial[m, k]*k!, {k, 0, m}], {m, 0, n}], {n, 0, 12}]
Formula
a(n) ~ c * exp(n+1) * BarnesG(n+2).
a(n) ~ c * n^(n^2/2 + n + 5/12) * (2*Pi)^(n/2 + 1/2) / (A * exp(3*n^2/4 - 13/12))
where c = 0.24314714161123874545254157058990661627416712475691705561000082745...
and A is the Glaisher-Kinkelin constant A074962.