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A228959
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Total sum of squared lengths of ascending runs in all permutations of [n].
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3
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0, 1, 6, 32, 186, 1222, 9086, 75882, 705298, 7231862, 81160422, 990024466, 13047411482, 184788881838, 2799459801742, 45178128866282, 773829771302946, 14021761172671462, 267991492197471158, 5388234382450264002, 113692608262971520042, 2512031106415692960926
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: (2*exp(x)-x-2)/(x-1)^2.
a(n) = (2*n+1)*a(n-1)-(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)) for n>=3, a(n) = n*(2*n-1) for n<3.
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EXAMPLE
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a(0) = 0: ().
a(1) = 1: (1).
a(2) = 6 = 4+2: (1,2), (2,1).
a(3) = 32 = 9+5+5+5+5+3: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
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MAPLE
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a:= proc(n) option remember; `if`(n<3, n*(2*n-1),
(2*n+1)*a(n-1) -(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)))
end:
seq(a(n), n=0..30);
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MATHEMATICA
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a[n_] := With[{k = 2}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
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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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