Search: id:A024166
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%I A024166
%S A024166 0,1,10,46,146,371,812,1596,2892,4917,7942,12298,18382,26663,37688,52088,
%T A024166 70584,93993,123234,159334,203434,256795,320804,396980,486980,592605,715806,
%U A024166 858690,1023526,1212751,1428976,1674992,1953776,2268497,2622522,3019422
%N A024166 Sum of (j-i)^3 for 1 <= i < j <= n.
%C A024166 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
%D A024166 La Recherche, April 1999, No. 319, page 97.
%D A024166 Alexander R. Povolotsky, www.pme-math.org/journal/ProblemsF2006.pdf and 
               http://www.math.fau.edu/web/PiMuEpsilon/pmespring2007.pdf
%H A024166 T. D. Noe, <a href="b024166.txt">Table of n, a(n) for n=0..1000</a>
%H A024166 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A024166 A. F. Labossiere, <a href="http://membres.lycos.fr/labos2/">New Artefact 
               From Pascal's Triangle</a>.
%H A024166 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               ">Miscellaneous</a>.
%F A024166 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
%F A024166 0, 0, 1, 10, ... has a(n)=sum_{k=0..n} k^3*(n-k) - Paul Barry (pbarry(AT)wit.ie), 
               Sep 14 2003
%F A024166 a(n) = Sum_{i=1..n} C(i+1, 2)^2. - Andre F. Labossiere (boronali(AT)laposte.net), 
               Jul 03 2003
%F A024166 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
%F A024166 Partial sums of A000537. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), 
               Dec 15 2004
%F A024166 Second partial sums of cubes (A000537).
%F A024166 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
%F A024166 O.g.f.: x*(1+4*x+x^2)/(-1+x)^6 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jun 06 2008
%t A024166 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]
%o A024166 (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
%Y A024166 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.
%Y A024166 Cf. A101094.
%Y A024166 Cf. A000330, A001286, A101102, A101097, A101094, A000537.
%Y A024166 Cf. A003215 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 20 
               2008]
%Y A024166 Sequence in context: A081583 A106600 A085437 this_sequence A103501 A003197 
               A096045
%Y A024166 Adjacent sequences: A024163 A024164 A024165 this_sequence A024167 A024168 
               A024169
%K A024166 nonn,easy,nice
%O A024166 0,3
%A A024166 Clark Kimberling (ck6(AT)evansville.edu)

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