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A051925
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n(2n+5)(n-1)/6.
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14
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Related to variance of number of inversions of a random permutation of n letters.
Zero followed by partial sums of A005563. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 17 2008]
Definition: A051925=A000330-A000027 (square pyramidal numbers minus natural numbers) [From Andrey Kostenko (Andrey.Kostenko(AT)buseco.monash.edu.au), Nov 30 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. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 15 2010]
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REFERENCES
| V. N. Sachkov, Probablistic Methods in Combinatorial Analysis, Cambridge, 1997.
J. Wang and H. Li, The upper bound of essential chromatic numbers of hypergraphs, Discr. Math. 254 (2002), 555-564.
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MAPLE
| a:=n->sum((n+j^2), j=0..n): seq(a(n), n=-1..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 27 2006
seq(sum(k^2-1, k=1..n), n=0..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 28 2008
with(finance):seq(add(cashflows([n, k^2, 0], 0 ), k=0..n), n=-1..45); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]
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MATHEMATICA
| lst={0}; s=0; Do[s+=n^2-1; AppendTo[lst, s], {n, 5!}]; lst...and/or... lst={}; Do[s=n*(2*n+5)*(n-1)/6; AppendTo[lst, s], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 30 2008]
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 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 15 2010]
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PROG
| (PARI) {print1(a=0, ", "); for(n=0, 42, print1(a=a+(n+1)^2-1, ", "))} [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 17 2008]
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CROSSREFS
| Sequence in context: A124078 A096795 A160039 * A011942 A101612 A123928
Adjacent sequences: A051922 A051923 A051924 * A051926 A051927 A051928
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 19 1999
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