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A051925 a(n) = n*(2*n+5)*(n-1)/6. 24
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Related to variance of number of inversions of a random permutation of n letters.
Zero followed by partial sums of A005563. - Klaus Brockhaus, Oct 17 2008
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
The sequence is related to A033487 by A033487(n-1) = n*a(n) - Sum_{i=0..n-1} a(i) = n*(n+1)*(n+2)*(n+3)/4. - Bruno Berselli, Apr 04 2012
Deleting the two 0's leaves row 2 of the convolution array A213750. - Clark Kimberling, Jun 20 2012
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
REFERENCES
V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.
LINKS
J. Wang and H. Li, The upper bound of essential chromatic numbers of hypergraphs, Discr. Math. 254 (2002), 555-564.
FORMULA
a(n) = A000330(n) - n. - Andrey Kostenko, Nov 30 2008
G.f.: x^2*(3-x)/(1-x)^4. - Colin Barker, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Apr 27 2012
E.g.f.: (x^2/6)*(2*x + 9)*exp(x). - G. C. Greubel, Jul 19 2017
From Amiram Eldar, Nov 10 2023: (Start)
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A362176 A160039 A272296 * A211811 A011942 A220147
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 19 1999
STATUS
approved

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Last modified July 12 19:36 EDT 2024. Contains 374252 sequences. (Running on oeis4.)