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

Author

Luciano Ancora, Feb 21 2015

Keywords

• nonn
• easy

Examples

			Second differences:  1, 126, 1932, 12138,  47544, ... (this sequence)
First differences:   1, 127, 2060, 14324,  63801, ... (A152726)
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The seventh powers:  1, 128, 2187, 16384,  78125, ... (A001015)
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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).

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254684, A255178, A255179.

Programs

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

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.

a(n) = A022523(n)-A022523(n-1). - R. J. Mathar, Jul 16 2015

Extensions

Edited by Bruno Berselli, Mar 19 2015