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A051925
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a(n) = n*(2*n+5)*(n-1)/6.
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23
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0, 0, 3, 11, 26, 50, 85, 133, 196, 276, 375, 495, 638, 806, 1001, 1225, 1480, 1768, 2091, 2451, 2850, 3290, 3773, 4301, 4876, 5500, 6175, 6903, 7686, 8526, 9425, 10385, 11408, 12496, 13651, 14875, 16170, 17538, 18981, 20501, 22100, 23780
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OFFSET
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0,3
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COMMENTS
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Related to variance of number of inversions of a random permutation of n letters.
a(n)/12 is the variance of the number of inversions of a random permutation of n letters. See evidence in Mathematica code below. - Geoffrey Critzer, May 15 2010
For n>=4, a(n-2) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0110 (the first n-4 zeros), or, the same, a(n-2) is up-down coefficient {n,6} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014
Minimum sum of the bottom row of a triangular array A filled with the integers [0..binomial(n, 2) - 1] that obeys the rule A[i, j] + 1 <= A[i+1, j] and A[i, j] + 1 <= A[i, j-1]. - C.S. Elder, Oct 13 2023
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REFERENCES
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V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.
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LINKS
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Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes II, arXiv:2407.08158 [math.CO], 2024. See p. 15.
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FORMULA
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Sum_{n>=2} 1/a(n) = 62/1225 + 24*log(2)/35.
Sum_{n>=2} (-1)^n/a(n) = 6*Pi/35 + 72*log(2)/35 - 2078/1225. (End)
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MATHEMATICA
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f[{x_, y_}] := 2 y - x^2; Table[f[Coefficient[ Series[Product[Sum[Exp[i t], {i, 0, m}], {m, 1, n - 1}]/n!, {t, 0, 2}], t, {1, 2}]], {n, 0, 41}]*12 (* Geoffrey Critzer, May 15 2010 *)
CoefficientList[Series[x^2*(3-x)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 27 2012 *)
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PROG
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(PARI) {print1(a=0, ", "); for(n=0, 42, print1(a=a+(n+1)^2-1, ", "))} \\ Klaus Brockhaus, Oct 17 2008
(Magma) I:=[0, 0, 3, 11]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 27 2012
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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