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
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
- Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Archive 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).
Programs
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Magma
[Binomial(n+5,6)*(3*n^2+15*n+10)/28: n in [1..30]]; // G. C. Greubel, Dec 01 2018
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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 *) -
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
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PARI
Vec(-x*(x^2+4*x+1)/(x-1)^9 + O(x^100)) \\ Colin Barker, Apr 23 2015
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PARI
a(n) = binomial(n+5,6)*(3*n^2+15*n+10)/28 \\ Charles R Greathouse IV, Apr 23 2015
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Sage
[binomial(n+5,6)*(3*n^2+15*n+10)/28 for n in (1..30)] # G. C. Greubel, Dec 01 2018
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
Extensions
Edited by Ralf Stephan, Dec 16 2004