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%I #124 Aug 31 2022 07:21:54
%S 0,1,10,46,146,371,812,1596,2892,4917,7942,12298,18382,26663,37688,
%T 52088,70584,93993,123234,159334,203434,256795,320804,396980,486980,
%U 592605,715806,858690,1023526,1212751,1428976,1674992,1953776,2268497,2622522,3019422
%N a(n) = Sum_{1 <= i < j <= n} (j-i)^3.
%C Convolution of the cubes (A000578) with the positive integers a(n)=n+1, where all sequences have offset zero. - _Graeme McRae_, Jun 06 2006
%C a(A004772(n)) mod 2 = 0; a(A016813(n)) mod 2 = 1. - _Reinhard Zumkeller_, Oct 14 2001
%C Partial sums of A000537. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%C a(n) gives the n-th antidiagonal sum of the convolution array A212891. - _Clark Kimberling_, Jun 16 2012
%C In general, the r-th successive summation of the cubes from 1 to n is (6*n^2 + 6*n*r + r^2 - r)*(n+r)!/((r+3)!*(n-1)!), n>0. Here r = 2. - _Gary Detlefs_, Mar 01 2013
%C The inverse binomial transform is (essentially) row n=2 of A087127. - _R. J. Mathar_, Aug 31 2022
%D Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.
%H T. D. Noe, <a href="/A024166/b024166.txt">Table of n, a(n) for n = 0..1000</a>
%H C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014
%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7
%H Alexander R. Povolotsky, <a href="http://www.pme-math.org/journal/ProblemsF2006.pdf">Problem 1147</a>, Pi Mu Epsilon Fall 2006 Problems.
%H Alexander R. Povolotsky, <a href="http://www.pme-math.org/journal/ProblemsS2007.pdf">Problem</a>, Pi Mu Epsilon Spring 2007 Problems.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)
%F From Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999: (Start)
%F a(n) = Sum_{i=0..n} (A000217(i))^2.
%F a(n) = n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60. (End)
%F a(n) = Sum_{k=0..n} k^3*(n+1-k). - _Paul Barry_, Sep 14 2003; edited by _Jon E. Schoenfield_, Dec 29 2014
%F a(n) = Sum_{i=1..n} binomial(i+1, 2)^2. - _André F. Labossière_, Jul 03 2003
%F 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_, May 17 2008
%F O.g.f.: x*(1 + 4*x + x^2)/(-1 + x)^6. - _R. J. Mathar_, Jun 06 2008
%F a(n) = (6*n^2 + 12*n + 2)*(n+2)!/(120*(n-1)!), n > 0. - _Gary Detlefs_, Mar 01 2013
%F a(n) = A222716(n+1)/10 = A000292(n)*A100536(n+1)/10. - _Jonathan Sondow_, Mar 04 2013
%F 4*a(n) = Sum_{i=0..n} A000290(i)*A000290(i+1). - _Bruno Berselli_, Feb 05 2014
%F a(n) = Sum_{i=1..n} Sum_{j=1..n} i*j*(n - max(i, j) + 1). - _Melvin Peralta_, May 12 2016
%F a(n) = n*binomial(n+3, 4) + binomial(n+2, 5). - _Tony Foster III_, Nov 14 2017
%F a(n) = Sum_{i=1..n} i*A143037(n,n-i+1). - _J. M. Bergot_, Aug 30 2022
%e 4*a(7) = 6384 = (0*1)^2 + (1*2)^2 + (2*3)^2 + (3*4)^2 + (4*5)^2 + (5*6)^2 + (6*7)^2 + (7*8)^2. - _Bruno Berselli_, Feb 05 2014
%p A024166:=n->n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60: seq(A024166(n), n=0..50); # _Wesley Ivan Hurt_, Nov 21 2017
%t Nest[Accumulate,Range[0,40]^3,2] (* _Harvey P. Dale_, Jan 10 2016 *)
%t Table[n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60, {n,0,30}] (* _G. C. Greubel_, Nov 21 2017 *)
%o (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_, May 17 2008
%o (Haskell)
%o a024166 n = sum $ zipWith (*) [n+1,n..0] a000578_list
%o -- _Reinhard Zumkeller_, Oct 14 2001
%o (Magma) [n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60: n in [0..30]]; // _G. C. Greubel_, Nov 21 2017
%o (PARI) for(n=0,30, print1(n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60, ", ")) \\ _G. C. Greubel_, Nov 21 2017
%Y Cf. A000292, A000332, A000389, A000579, A000580, A024166, A027555, A085438, A085439, A085440, A085441, A085442, A086020, A086021, A086022, A086023, A086024, A086025, A086026, A086027, A086028, A086029, A086030, A087127.
%Y Cf. A000330, A000537 (first diffs), A001286, A003215, A100536, A101094 (partial sums), A101097, A101102, A222716.
%K nonn,easy,nice
%O 0,3
%A _Clark Kimberling_
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