A254684 Fifth partial sums of seventh powers (A001015).
1, 133, 2842, 29274, 197400, 1001952, 4137966, 14597934, 45454773, 127861825, 330540028, 795609724, 1801339176, 3867558072, 7926516900, 15591322404, 29566276257, 54259095093, 96674782246, 167695627750, 283882296880
Offset: 1
Examples
First differences: 1, 127, 2060, 14324, 63801, ... (A152726) ---------------------------------------------------------------------- The seventh powers: 1, 128, 2187, 16384, 78125, ... (A001015) ---------------------------------------------------------------------- First partial sums: 1, 129, 2316, 18700, 96825, ... (A000541) Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212) Third partial sums: 1, 131, 2577, 23723, 141694, ... (A254641) Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (A254646) Fifth partial sums: 1, 133, 2842, 29274, 197400, ... (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
- Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
Programs
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Mathematica
Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 3 + 5 n + n^2) (- 2 + 5 n + n^2) (5 + 5 n + n^2)/95040, {n,21}] (* or *) CoefficientList[Series[(- 1 - 120 x - 1191 x^2 - 2416 x^3 - 1191 x^4 - 120 x^5 - x^6)/(-1 + x)^13, {x,0,20}], x] -
PARI
a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-3+5*n+n^2)*(-2+5*n+n^2)*(5+5*n+n^2)/95040 \\ Charles R Greathouse IV, Oct 07 2015
Formula
G.f.: (- x - 120*x^2 - 1191*x^3 - 2416*x^4 - 1191*x^5 - 120*x^6 - x^7)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(-3 + 5*n + n^2)*(-2 + 5*n + n^2)*(5 + 5*n + n^2)/95040.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^7.