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

0,2

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).

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

Example

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)

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]

Prog

(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
(Python)
def A255177(n): return 14*n*(n**2*(3*n**2 + 5) + 1) if n else 1 # Chai Wah Wu, Oct 07 2024

Crossrefs

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

Sequence in context: A267628 A304559 A189345 * A172140 A331100 A196414

Adjacent sequences: A255174 A255175 A255176 * A255178 A255179 A255180

Keyword

nonn,easy

Author

Luciano Ancora, Feb 21 2015

Extensions

Edited by Bruno Berselli, Mar 19 2015

Status

approved