login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A024166 a(n) = Sum_{1 <= i < j <= n} (j-i)^3. 48
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; text; internal format)
OFFSET

0,3

COMMENTS

Convolution of the cubes (A000578) with the positive integers a(n)=n+1, where all sequences have offset zero. - Graeme McRae, Jun 06 2006

a(A004772(n)) mod 2 = 0; a(A016813(n)) mod 2 = 1. - Reinhard Zumkeller, Oct 14 2001

a(n) gives the n-th antidiagonal sum of the convolution array A212891. - Clark Kimberling, Jun 16 2012

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

REFERENCES

La Recherche, April 1999, No. 319, page 97.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014

Alexander R. Povolotsky, Problem 1147, Pi Mu Epsilon Fall 2006 Problems.

Alexander R. Povolotsky, Problem, Pi Mu Epsilon Spring 2007 Problems.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

FORMULA

a(n) = Sum_{i=0..n} (A000217(i))^2 = n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

a(n) = Sum_{k=0..n} k^3*(n+1-k). - Paul Barry, Sep 14 2003; edited by Jon E. Schoenfield, Dec 29 2014

a(n) = Sum_{i=1..n} binomial(i+1, 2)^2. - André F. Labossière, Jul 03 2003

Partial sums of A000537. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

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

O.g.f.: x*(1 + 4*x + x^2)/(-1 + x)^6 . - R. J. Mathar, Jun 06 2008

a(n) = (6*n^2 + 12*n + 2)*(n+2)!/(120*(n-1)!), n > 0. - Gary Detlefs, Mar 01 2013

a(n) = A222716(n+1)/10 = A000292(n)*A100536(n+1)/10. - Jonathan Sondow, Mar 04 2013

4*a(n) = Sum_{i=0..n} A000290(i)*A000290(i+1). - Bruno Berselli, Feb 05 2014

a(n) = Sum_{i=1..n} Sum_{j=1..n} i*j*(n - max(i, j) + 1) - Melvin Peralta, May 12 2016

EXAMPLE

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

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 (* Vladimir Joseph Stephan Orlovsky, Dec 20 2008 *)

Nest[Accumulate, Range[0, 40]^3, 2] (* Harvey P. Dale, Jan 10 2016 *)

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, May 17 2008

(Haskell)

a024166 n = sum $ zipWith (*) [n+1, n..0] a000578_list

-- Reinhard Zumkeller, Oct 14 2001

CROSSREFS

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.

Cf. A000330, A000537, A001286, A003215, A100536, A101094, A101097, A101102, A222716.

Sequence in context: A241084 A106600 A085437 * A103501 A219003 A003197

Adjacent sequences:  A024163 A024164 A024165 * A024167 A024168 A024169

KEYWORD

nonn,easy,nice

AUTHOR

Clark Kimberling

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 22 12:49 EDT 2017. Contains 290947 sequences.