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A024166
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Sum of (j-i)^3 for 1 <= i < j <= n.
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34
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0, 1, 10, 46, 146, 371, 812, 1596, 2892, 4917, 7942, 12298, 18382, 26663, 37688, 52088, 70584, 93993, 123234, 159334, 203434, 256795, 320804, 396980, 486980, 592605, 715806, 858690, 1023526, 1212751, 1428976, 1674992, 1953776, 2268497, 2622522, 3019422
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Convolution of the cubes (A000578) with the positive integers a(n)=n+1, where all sequences have offset zero. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 06 2006
a(A004772(n)) mod 2 = 0; a(A016813(n)) mod 2 = 1. [Reinhard Zumkeller, Oct 14 2001]
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REFERENCES
| La Recherche, April 1999, No. 319, page 97.
Alexander R. Povolotsky, www.pme-math.org/journal/ProblemsF2006.pdf and http://www.math.fau.edu/web/PiMuEpsilon/pmespring2007.pdf
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n)=sum((A000217(i))^2, i=0..n) = (1/60)*n*(n+1)*(n+2)*(3*n^2+6*n+1) - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
0, 0, 1, 10, ... has a(n)=sum_{k=0..n} k^3*(n-k) - Paul Barry (pbarry(AT)wit.ie), Sep 14 2003
a(n) = Sum_{i=1..n} C(i+1, 2)^2. - Andre F. Labossiere (boronali(AT)laposte.net), Jul 03 2003
a(n) = ( 6*(n^5) + 30*(n^4) + 50*(n^3) + 30*(n^2) + 4*n )/5!. - Andre F. Labossiere (boronali(AT)laposte.net), Jul 03 2003
Partial sums of A000537. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
Second partial sums of cubes (A000537).
a(n)= 2*n*(n+1)*(n+2)*((n+1)^2 + 2*n*(n+2))/5!. This sequence could be obtained from the general formula a(n)=n*(n+1)*(n+2)*(n+3)* ...* (n+k) *(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=2 - Alexander R. Povolotsky (pevnev(AT)juno.com), May 17 2008
O.g.f.: x*(1+4*x+x^2)/(-1+x)^6 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 06 2008
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MATHEMATICA
| c[n_]:=n^3; d[n_]:=Sum[c[i], {i, 0, n}]; e[n_]:=Sum[d[i], {i, 0, n}]; lst={}; Do[AppendTo[lst, e[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 20 2008]
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PROG
| (PARI) a(n)=sum(j=1, n, sum(m=1, j, sum(i=m*(m+1)/2-m+1, m*(m+1)/2, (2*i-1)))) - Alexander R. Povolotsky (pevnev(AT)juno.com), May 17 2008
(Haskell)
a024166 n = sum $ zipWith (*) [n+1, n..0] a000578_list
-- Reinhard Zumkeller, Oct 14 2001
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CROSSREFS
| Cf. A000292, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.
Cf. A101094.
Cf. A000330, A001286, A101102, A101097, A101094, A000537.
Cf. A003215 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 20 2008]
Sequence in context: A081583 A106600 A085437 * A103501 A003197 A096045
Adjacent sequences: A024163 A024164 A024165 * A024167 A024168 A024169
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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