A254682 Fifth partial sums of fifth powers (A000584).
1, 37, 418, 2754, 13080, 49632, 159654, 452166, 1157013, 2724865, 5988268, 12410476, 24456744, 46132152, 83740980, 146935284, 250134753, 414416277, 669990046, 1059399550, 1641605680, 2497140360, 3734542890, 5498322570
Offset: 1
Examples
Fifth differences: 1, 27, 93, 119, 120, (repeat 120) (A101100) Fourth differences: 1, 28, 121, 240, 360, 480, ... (A101095) Third differences: 1, 29, 150, 390, 750, 1230, ... (A101096) Second differences: 1, 30, 180, 570, 1320, 2550, ... (A101098) First differences: 1, 31, 211, 781, 2101, 4651, ... (A022521) ------------------------------------------------------------------------- The fifth powers: 1, 32, 243, 1024, 3125, 7776, ... (A000584) ------------------------------------------------------------------------- First partial sums: 1, 33, 276, 1300, 4425, 12201, ... (A000539) Second partial sums: 1, 34, 310, 1610, 6035, 18236, ... (A101092) Third partial sums: 1, 35, 345, 1955, 7990, 26226, ... (A101099) Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (A254644) Fifth partial sums: 1, 37, 418, 2754, 13080, 49632, ... (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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).
Crossrefs
Programs
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Mathematica
Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 2 + 5 n + n^2) (9 + 10 n + 2 n^2)/60480, {n,24}] (* or *) CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^11, {x,0,23}], x] Nest[Accumulate,Range[30]^5,5] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,37,418,2754,13080,49632,159654,452166,1157013,2724865,5988268},30] (* Harvey P. Dale, Jan 30 2019 *) -
PARI
a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-2+5*n+n^2)*(9+10*n+2*n^2)/60480 \\ Charles R Greathouse IV, Oct 07 2015
Formula
G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(- 2 + 5*n + n^2)*(9 + 10*n + 2*n^2)/60480.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^5.
Sum_{n>=1} 1/a(n) = 475867/180 - (2560/13)*sqrt(7)*Pi*tan(sqrt(7)*Pi/2) + (210/13)*sqrt(3/11)*Pi*tan(sqrt(33)*Pi/2). - Amiram Eldar, Jan 27 2022