A254869 Seventh partial sums of cubes (A000578).
1, 15, 111, 561, 2211, 7293, 21021, 54483, 129558, 286858, 598026, 1184118, 2242266, 4083366, 7184166, 12257850, 20348031, 32951985, 52179985, 80958735, 123288165, 184562235, 271965915, 394962165, 565884540, 800652996, 1119632580, 1548656956
Offset: 1
Examples
2nd differences: 0, 6, 12, 18, 24, 30, ... (A008588) 1st differences: 1, 7, 19, 37, 61, 91, ... (A003215) ------------------------------------------------------------------- The cubes: 1, 8, 27, 64, 125, 216, ... (A000578) ------------------------------------------------------------------- 1st partial sums: 1, 9, 36, 100, 225, 441, ... (A000537) 2nd partial sums: 1, 10, 46, 146, 371, 812, ... (A024166) 3rd partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094) 4th partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097) 5th partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102) 6th partial sums: 1, 14, 96, 450, 1650, 5082, ... (A254469) 7th partial sums: 1, 15, 111, 561, 2211, 7293, ... (this sequence)
Links
- Luciano Ancora, Table of n, a(n) for n = 1..1000
- Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials.
- Luciano Ancora, Pascal's triangle and recurrence relations for partial sums of m-th powers.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
Crossrefs
Programs
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Magma
[n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+7*n+n^2)/604800: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015
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Mathematica
Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 7 n + n^2)/604800, {n, 26}] (* or *) CoefficientList[Series[(- 1 - 4 x - x^2)/(- 1 + x)^11, {x, 0, 25}], x] Nest[Accumulate,Range[30]^3,7] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,15,111,561,2211,7293,21021,54483,129558,286858,598026},30] (* Harvey P. Dale, Apr 24 2017 *) -
PARI
vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800) \\ Derek Orr, Feb 19 2015
Formula
G.f.: x*(1 + 4*x + x^2)/(1 - x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^3.
Sum_{n>=1} 1/a(n) = 1920*sqrt(3/7)*Pi*tan(sqrt(21)*Pi/2) - 251488/49. - Amiram Eldar, Jan 26 2022