A254644 Fourth partial sums of fifth powers (A000584).
1, 36, 381, 2336, 10326, 36552, 110022, 292512, 704847, 1567852, 3263403, 6422208, 12046268, 21675408, 37608828, 63194304, 103199469, 164281524, 255573769, 389409504, 582206130, 855534680, 1237402530, 1763779680, 2480401755, 3444885756, 4729197591, 6422513536, 8634521016, 11499207456
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, ... (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(10,-45,120,-210,252,-210,120,-45,10,-1).
Crossrefs
Programs
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GAP
List([1..30], n-> Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126); # G. C. Greubel, Aug 28 2019
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Magma
[Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126: n in [1..30]]; // G. C. Greubel, Aug 28 2019
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Maple
seq(binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126, n=1..30); # G. C. Greubel, Aug 28 2019
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Mathematica
Table[n(1+n)(2+n)(3+n)(4+n)(-24 +20n +85n^2 +40n^3 +5n^4)/15120, {n, 30}] (* or *) Accumulate[Accumulate[Accumulate[Accumulate[Range[24]^5]]]] (* or *) CoefficientList[Series[(1 +26x +66x^2 +26x^3 +x^4)/(1-x)^10, {x, 0, 30}], x] Nest[Accumulate,Range[30]^5,4] (* or *) LinearRecurrence[{10,-45,120, -210,252,-210,120,-45,10,-1}, {1,36,381,2336,10326,36552,110022,292512, 704847,1567852},30] (* Harvey P. Dale, May 08 2016 *) -
PARI
vector(30, n, m=n+2; binomial(m+2,5)*(5*m^4 -35*m^2 +36)/126) \\ G. C. Greubel, Aug 28 2019
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Sage
[binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126 for n in (1..30)] # G. C. Greubel, Aug 28 2019
Formula
G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^10.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(-24 + 20*n + 85*n^2 + 40*n^3 + 5*n^4)/15120.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^5.
Extensions
Edited by Bruno Berselli, Feb 10 2015