A344054 a(n) = Sum_{k = 0..n} E1(n, k)*k^2, where E1 are the Eulerian numbers A173018.

0, 0, 1, 8, 64, 540, 4920, 48720, 524160, 6108480, 76809600, 1037836800, 15008716800, 231437606400, 3792255667200, 65819609856000, 1206547550208000, 23297526540288000, 472708591939584000, 10055994967130112000, 223826984752250880000, 5202760944485744640000, 126075414965721661440000, 3179798058882852126720000, 83346901966165164687360000, 2267221868000212451328000000

Offset

0,4

Comments

The Eulerian transform of the squares.

Links

Table of n, a(n) for n=0..25.

Formula

a(n) = n! * [x^n] x^2*(-x^2 + x - 3)/(6*(x - 1)^3).

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(n + 1, j)*k^2*(k + 1 - j)^n.

a(n) = ((n - 3)*(n - 1)*(23*n - 44)*a(n-2) + ((159 - 7*n)*n - 286)*a(n-1))/(16*(n - 2)) for n >= 3.

Maple

a := n -> add(combinat[eulerian1](n, k)*k^2, k = 0..n):
# Recurrence:
a := proc(n) option remember; if n < 2 then 0 elif n = 2 then 1 else
((n-3)*(n-1)*(23*n-44)*a(n-2) + ((159 - 7*n)*n - 286)*a(n-1))/(16*(n - 2)) fi end:
seq(a(n), n = 0..29);

Mathematica

a[n_] := Sum[Sum[(-1)^j Binomial[n + 1, j] k^2 (k + 1 - j)^n, {j, 0, k}], {k, 0, n}]; a[0] := 0; Table[a[n], {n, 0, 25}]

Prog

(SageMath)
def aList(len):
    R.<x> = PowerSeriesRing(QQ, default_prec=len+2)
    f = x^2*(-x^2 + x - 3)/(6*(x - 1)^3)
    return f.egf_to_ogf().list()[:len]
print(aList(20))

Crossrefs

Transforms of the squares: A151881 (StirlingCycle), A033452 (StirlingSet), A105219 (Laguerre), A103194 (Lah), A065096 (SchröderBig), A083411 (Fubini), A141222 (Narayana), A000330 (Units A000012).

Cf. A173018.

Keyword

nonn

Author

Peter Luschny, May 11 2021

Status

approved