A294353 -id:A294353 - cp's OEIS frontend

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

Vaclav Kotesovec, Oct 29 2017

nonn

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.

Cf. A000522, A047656, A086619, A294353.

Table[Product[Sum[Binomial[m, k]*k!, {k, 0, m}], {m, 0, n}], {n, 0, 12}]

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.

A110140 Binomial transform of n^n (with interpolated zeros).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 103, 351, 1321, 5113, 21201, 90465, 406621, 1879021, 9051309, 44754061, 229059633, 1201271409, 6488957521, 35853950609, 203303964341, 1177299817093, 6976709899853, 42161309544541, 260154190137865

Offset: 0

Paul Barry, Jul 13 2005

Binomial transform of 1,0,1,0,4,0,27,....

Cf. A000312, A086331, A294353.

Table[1 + Sum[Binomial[n, k]*(k/2)^(k/2)*(1 + (-1)^k)/2, {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 30 2017 *)

a(n)=sum{k=0..n, C(n, k)(k/2)^(k/2)*(1+(-1)^k)/2}

cp's OEIS Frontend

A294352 Product of first n terms of the binomial transform of the factorial.

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