OFFSET
0,2
COMMENTS
a(n) is the first entry in the n-th row of a triangular array built from the factorial sequence, where T(0, m) = m! and each subsequent row T(n, m) is obtained by T(n - 1, m + 3) - 2*T(n - 1, m + 2) + T(n - 1, m + 1).
The rows are generated by iterating the operator E*Delta^2, where E is the forward shift operator, E(f(m)) = f(m + 1), and Delta = E - 1 is the forward difference operator. Equivalently, E*Delta^2 = E*(E - 1)^2 = E^3 - 2*E^2 + E.
This is the (p, q) = (2, 1) case of the family obtained by iterating E^q*Delta^p on the factorial sequence.
FORMULA
Let f(m) = m!. Then
a(n) = ((E*Delta^2)^n f)(0).
a(n) = Sum_{j=0..2*n} (-1)^j*binomial(2*n, j)*(3*n - j)!.
a(n) = Integral_{t=0..oo} t^n*(t - 1)^(2*n)*exp(-t) dt.
More generally, if b_{p,q}(n) = ((E^q*Delta^p)^n f)(0), where f(m) = m!, then b_{p,q}(n) = Integral_{t=0..oo} t^(q*n)*(t - 1)^(p*n)*exp(-t) dt = Sum_{j=0..p*n} (-1)^j*binomial(p*n, j)*((p + q)*n - j)!.
a(n) ~ sqrt(2*Pi) * 3^(3*n + 1/2) * n^(3*n + 1/2) / exp(3*n + 2/3). - Vaclav Kotesovec, May 13 2026
EXAMPLE
First column of
n=0: 1 1 2 6 24 120 720 5040
n=1: 3 14 78 504 3720 30960 287280
n=2: 362 2790 24024 229080 2399760 27422640
n=3: 183822 1965624 22852200 287250480 3884393520
n=4: 243511704 3332744760 48773612880 760381337520
MATHEMATICA
Table[Sum[(-1)^j * Binomial[2*n, j] * (3*n - j)!, {j, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, May 13 2026 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Dalton Heilig, May 05 2026
STATUS
approved