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A052754
Expansion of e.g.f.: (log(1-x))^2*x^2.
4
0, 0, 0, 0, 24, 120, 660, 4200, 30688, 254016, 2352240, 24108480, 271016064, 3316135680, 43877957760, 624306009600, 9505324339200, 154205312163840, 2655567756979200, 48382249157222400, 929788248840192000, 18796669969158144000, 398766195659497881600
OFFSET
0,5
COMMENTS
Previous name was: A simple grammar.
LINKS
FORMULA
E.g.f.: log(-1/(-1+x))^2*x^2.
Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=24, (n^4-6*n^2-n^3+4*n+8)*a(n) + (7*n-2*n^3+n^2-6)*a(n+1) + (n^2-n)*a(n+2) = 0.
a(n) ~ (n-1)! * 2*(log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 01 2013
a(n) = n*A052745(n-1) = 2*n*(n-1)*abs(Stirling1(n-2,2)) for n >= 2. - Andrew Howroyd, Aug 08 2020
MAPLE
spec := [S, {B=Cycle(Z), S=Prod(B, B, Z, Z)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
CoefficientList[Series[(Log[1-x])^2*x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
Join[{0, 0, 0, 0}, RecurrenceTable[{a[4] == 24, a[5] == 120, (n^4 - 6*n^2 - n^3 + 4*n + 8)*a[n] + (7*n - 2*n^3 + n^2 - 6)*a[n + 1] == -(n^2 - n)*a[n + 2]}, a, {n, 4, 30}]] (* G. C. Greubel, Sep 05 2018 *)
PROG
(PARI) x='x+O('x^30); concat(vector(4), Vec(serlaplace(log(-1/(-1+x))^2* x^2))) \\ G. C. Greubel, Sep 05 2018
(PARI) a(n)={if(n>=2, 2*n*(n-1)*abs(stirling(n-2, 2, 1)), 0)} \\ Andrew Howroyd, Aug 08 2020
(Magma) I:=[24, 120]; [0, 0, 0, 0] cat [n le 2 select I[n] else (n*(n+3)*(2*n-1)*Self(n-1) - (n-1)^2*(n+2)*(n+3)*Self(n-2))/(n*(n+1)): n in [1..30]]; // G. C. Greubel, Sep 05 2018
CROSSREFS
Sequence in context: A052760 A179720 A235702 * A050213 A124657 A342856
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New name using e.g.f., Vaclav Kotesovec, Oct 01 2013
STATUS
approved