A051925 a(n) = n*(2*n+5)*(n-1)/6.

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

Offset: 0

Author

N. J. A. Sloane, Dec 19 1999

Keywords

• nonn
• easy

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

The preceding statement can be extended: a(n) is the minimum sum of the main antidiagonal of a n X n square array A filled eith the integers [0..n^2-1] that is increasing on each row from left to right, and on each column from top to bottom. - Yifan Xie, Dec 19 2024

References

• V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Jianfang Wang and Haizhu Li, The upper bound of essential chromatic numbers of hypergraphs, Discr. Math. 254 (2002), 555-564.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000330, A005563, A033487.

Programs

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

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 *)
LinearRecurrence[{4,-6,4,-1},{0,0,3,11},50] (* Harvey P. Dale, Sep 07 2024 *)

PARI

{print1(a=0, ","); for(n=0, 42, print1(a=a+(n+1)^2-1, ","))} \\ Klaus Brockhaus, Oct 17 2008

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)