A395765 First column of the triangular array with T(0, m) = m! and T(n, m) = T(n - 1, m + 3) - T(n - 1, m + 2).

1, 4, 504, 256320, 339696000, 929459059200, 4557208289356800, 36403110891295948800, 442406862895153508352000, 7768652131253698010726400000, 189329460672170539873521008640000, 6200149639791767548556682032578560000, 265694561569328384410252384995689103360000

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 is obtained by T(n, m) = T(n - 1, m + 3) - T(n - 1, m + 2).

The rows are generated by iterating the operator E^2*Delta, where E is the forward shift operator, E(f(m)) = f(m + 1), and Delta = E - 1 is the forward difference operator. Equivalently, E^2*Delta = E^2*(E - 1) = E^3 - E^2.

This is the (p, q) = (1, 2) case of the family obtained by iterating E^q*Delta^p on the factorial sequence. The case (p, q) = (1, 0) gives the derangement numbers A000166 and the case (p, q) = (1, 1) gives A033815.

Links

Table of n, a(n) for n=0..12.

Formula

Let f(m) = m!. Then:

a(n) = ((E^2*Delta)^n f)(0).

a(n) = Sum_{j=0..n} (-1)^j*binomial(n, j)*(3*n - j)!.

a(n) = Integral_{t=0..oo} t^(2*n)*(t - 1)^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 + 1/3). - Vaclav Kotesovec, May 13 2026

Example

First column of
  row 0:         1          1           2            6          24       120     720  5040
  row 1:         4         18          96          600        4320     35280  322560
  row 2:       504       3720       30960       287280     2943360  33022080
  row 3:    256320    2656080    30078720    369774720  4906137600
  row 4: 339696000 4536362880 64988179200 994447238400

Mathematica

Table[Sum[(-1)^j * Binomial[n, j] * (3*n - j)!, {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 13 2026 *)

Crossrefs

Cf. A000142, A000166, A033815, A392154.

Sequence in context: A350968 A378310 A221550 * A022919 A253904 A268468

Adjacent sequences: A395762 A395763 A395764 * A395766 A395767 A395768

Keyword

nonn

Author

Dalton Heilig, May 05 2026

Status

approved