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A344054
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a(n) = Sum_{k = 0..n} E1(n, k)*k^2, where E1 are the Eulerian numbers A173018.
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0
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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
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OFFSET
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0,4
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COMMENTS
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The Eulerian transform of the squares.
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LINKS
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FORMULA
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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.
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MAPLE
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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);
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MATHEMATICA
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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}]
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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