A101102 Fifth partial sums of cubes (A000578).

1, 13, 82, 354, 1200, 3432, 8646, 19734, 41613, 82225, 153868, 274924, 472056, 782952, 1259700, 1972884, 3016497, 4513773, 6624046, 9550750, 13550680, 18944640, 26129610, 35592570, 47926125, 63846081, 84211128, 110044792, 142559824, 183185200, 233595912

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]

Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(10 + 3*n*(n+5))/20160.

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=5. - Alexander R. Povolotsky, May 17 2008

G.f.: x*(x^2+4*x+1) / (1-x)^9. - Colin Barker, Apr 23 2015

Sum_{n>=1} 1/a(n) = -162*sqrt(21/5)*Pi*tan(sqrt(35/3)*Pi/2) - 136269/100. - Amiram Eldar, Jan 26 2022

Mathematica

Table[Binomial[n+5, 6]*(3*n^2+15*n+10)/28, {n, 1, 30}] (* G. C. Greubel, Dec 01 2018 *)
Nest[Accumulate, Range[40]^3, 5] (* Harvey P. Dale, Feb 06 2023 *)

Prog

(PARI) a(n)=sum(t=1, n, sum(s=1, t, sum(l=1, s, sum(j=1, l, 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
(PARI) Vec(-x*(x^2+4*x+1)/(x-1)^9 + O(x^100)) \\ Colin Barker, Apr 23 2015
(PARI) a(n) = binomial(n+5, 6)*(3*n^2+15*n+10)/28 \\ Charles R Greathouse IV, Apr 23 2015
(Magma) [Binomial(n+5, 6)*(3*n^2+15*n+10)/28: n in [1..30]]; // G. C. Greubel, Dec 01 2018
(Sage) [binomial(n+5, 6)*(3*n^2+15*n+10)/28 for n in  (1..30)] # G. C. Greubel, Dec 01 2018

Crossrefs

Partial sums of A101097.

Cf. A000537, A024166, A101094.

Sequence in context: A362545 A082203 A367118 * A213572 A142085 A163688

Adjacent sequences: A101099 A101100 A101101 * A101103 A101104 A101105

Keyword

easy,nonn

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Extensions

Edited by Ralf Stephan, Dec 16 2004

Status

approved