A255179 Second differences of ninth powers (A001017).
1, 510, 18660, 223290, 1448520, 6433590, 22151340, 63588210, 159338640, 359376750, 745368180, 1443884970, 2642886360, 4611828390, 7725765180, 12493804770, 19592282400, 29903014110, 44556993540, 64983894810, 92967744360, 130709124630, 180894272460
Offset: 0
Examples
Second differences: 1, 510, 18660, 223290, 1448520, ... (this sequence) First differences: 1, 511, 19171, 242461, 1690981, ... (A022525) ------------------------------------------------------------------------ The ninth powers: 1, 512, 19683, 262144, 1953125, ... (A001017) ------------------------------------------------------------------------ First partial sums: 1, 513, 20196, 282340, 2235465, ... (A007487) Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637) Third partial sums: 1, 515, 21225, 324275, 2862790, ... (A254643)
Links
- Luciano Ancora, Table of n, a(n) for n = 0..1000
- Luciano Ancora, Sums of powers of positive integers and their recurrence relations, section 0.5.
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Programs
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Magma
[1] cat [6*n*(3+28*n^2+42*n^4+12*n^6): n in [1..30]]; // Vincenzo Librandi, Mar 12 2015
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Mathematica
Join[{1}, Table[6 n (3 + 28 n^2 + 42 n^4 + 12 n^6), {n, 1, 30}]] Join[{1},Differences[Range[0,30]^9,2]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,510,18660,223290,1448520,6433590,22151340,63588210,159338640},30] (* Harvey P. Dale, Jan 26 2019 *)
Formula
G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^8.
a(n) = 6*n*(3 + 28*n^2 + 42*n^4 + 12*n^6) for n>0, a(0)=1.
Extensions
Corrected g.f. by Bruno Berselli, Feb 25 2015
Offset changed by Bruno Berselli, Mar 20 2015