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%I #100 Nov 10 2023 09:24:11
%S 0,0,3,11,26,50,85,133,196,276,375,495,638,806,1001,1225,1480,1768,
%T 2091,2451,2850,3290,3773,4301,4876,5500,6175,6903,7686,8526,9425,
%U 10385,11408,12496,13651,14875,16170,17538,18981,20501,22100,23780
%N a(n) = n*(2*n+5)*(n-1)/6.
%C Related to variance of number of inversions of a random permutation of n letters.
%C Zero followed by partial sums of A005563. - _Klaus Brockhaus_, Oct 17 2008
%C 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
%C 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
%C Deleting the two 0's leaves row 2 of the convolution array A213750. - _Clark Kimberling_, Jun 20 2012
%C 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
%C 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
%D V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.
%H Vincenzo Librandi, <a href="/A051925/b051925.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Wang and H. Li, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00301-6">The upper bound of essential chromatic numbers of hypergraphs</a>, Discr. Math. 254 (2002), 555-564.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A000330(n) - n. - _Andrey Kostenko_, Nov 30 2008
%F G.f.: x^2*(3-x)/(1-x)^4. - _Colin Barker_, Apr 04 2012
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Apr 27 2012
%F E.g.f.: (x^2/6)*(2*x + 9)*exp(x). - _G. C. Greubel_, Jul 19 2017
%F From _Amiram Eldar_, Nov 10 2023: (Start)
%F Sum_{n>=2} 1/a(n) = 62/1225 + 24*log(2)/35.
%F Sum_{n>=2} (-1)^n/a(n) = 6*Pi/35 + 72*log(2)/35 - 2078/1225. (End)
%t 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 *)
%t CoefficientList[Series[x^2*(3-x)/(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Apr 27 2012 *)
%o (PARI) {print1(a=0, ","); for(n=0, 42, print1(a=a+(n+1)^2-1, ","))} \\ _Klaus Brockhaus_, Oct 17 2008
%o (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
%Y Cf. A000330, A005563, A033487.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 19 1999
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