A254469 Sixth partial sums of cubes (A000578).
1, 14, 96, 450, 1650, 5082, 13728, 33462, 75075, 157300, 311168, 586092, 1058148, 1841100, 3100800, 5073684, 8090181, 12603954, 19228000, 28778750, 42329430, 61274070, 87403680, 122996250, 170922375, 234768456, 318979584, 429024376, 571584200, 754769400
Offset: 1
Examples
First differences: 1, 7, 19, 37, 61, 91, ... (A003215) ------------------------------------------------------------------------- The cubes: 1, 8, 27, 64, 125, 216, ... (A000578) ------------------------------------------------------------------------- First partial sums: 1, 9, 36, 100, 225, 441, ... (A000537) Second partial sums: 1, 10, 46, 146, 371, 812, ... (A024166) Third partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094) Fourth partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097) Fifth partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102) Sixth partial sums: 1, 14, 96, 450, 1650, 5082, ... (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 (10,-45,120,-210,252,-210,120,-45,10,-1).
Crossrefs
Programs
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Magma
[n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
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Mathematica
Table[n (1 + n)^2 (2 + n) (3 + n) (4 + n) (5 + n)^2 (6 + n)/60480, {n, 27}] (* or *) CoefficientList[Series[(1 + 4 x + x^2)/(- 1 + x)^10, {x, 0, 26}], x] Nest[Accumulate,Range[30]^3,6] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,14,96,450,1650,5082,13728,33462,75075,157300},30] (* Harvey P. Dale, Sep 03 2016 *) -
PARI
a(n)=n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480 \\ Charles R Greathouse IV, Oct 07 2015
Formula
G.f.: (x + 4*x^2 + x^3)/(- 1 + x)^10.
a(n) = n*(1 + n)^2*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(6 + n)/60480.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^3.
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 217/200.
Sum_{n>=1} (-1)^(n+1)/a(n) = 223769/200 - 8064*log(2)/5. (End)