A255177 Second differences of seventh powers (A001015).
1, 126, 1932, 12138, 47544, 140070, 341796, 730002, 1412208, 2531214, 4270140, 6857466, 10572072, 15748278, 22780884, 32130210, 44327136, 59978142, 79770348, 104476554, 134960280, 172180806, 217198212, 271178418
Offset: 0
Examples
Second differences: 1, 126, 1932, 12138, 47544, ... (this sequence) 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, ... (A254684)
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 (6,-15,20,-15,6,-1).
Programs
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Magma
[1] cat [14*(-1+n)*(9-22*n+23*n^2-12*n^3+3*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 12 2015
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Mathematica
Join[{1}, Table[14 n (3 n^4 + 5 n^2 + 1), {n, 1, 30}], {n, 0, 24}] (* or *) CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(1 - x)^6, {x, 0, 22}], x] -
Python
def A255177(n): return 14*n*(n**2*(3*n**2 + 5) + 1) if n else 1 # Chai Wah Wu, Oct 07 2024
Formula
G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^6.
a(n) = 14*n*(3*n^4 + 5*n^2 + 1) for n>0, a(0)=1.
Extensions
Edited by Bruno Berselli, Mar 19 2015