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.

Sequence in context: A238300 A344271 A144317 * A344252 A199567 A213336

Adjacent sequences: A344051 A344052 A344053 * A344055 A344056 A344057

Keyword

nonn

Author

Peter Luschny, May 11 2021

Status

approved