A254642 Third partial sums of eighth powers (A001016).

1, 259, 7335, 86765, 629174, 3314178, 13906578, 49183590, 152191935, 422931613, 1075761505, 2540663307, 5633367740, 11829663860, 23692442292, 45516670332, 84278105421, 150996708135, 262656041515, 444856105561, 735419759634, 1189222877270

Offset

1,2

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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Formula

G.f.: (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)/(- 1 + x)^12.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(3 + 2*n)*(1 + 36*n - 69*n^2 + 45*n^4 + 18*n^5 + 2*n^6)/3960.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^8.

Example

First differences:   1, 255, 6305, 58975, 325089, ...(A022524)
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The eighth powers:   1, 256, 6561, 65536, 390625, ...(A001016)
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First partial sums:  1, 257, 6818, 72354, 462979, ...(A000542)
Second partial sums: 1, 258, 7076, 79430, 542409, ...(A253636)
Third partial sums:  1, 259, 7335, 86765, 629174, ...(this sequence)

Mathematica

Table[n (1 + n) (2 + n) (3 + n) (3 + 2 n) (1 + 36 n - 69 n^2 + 45 n^4 + 18 n^5 + 2 n^6)/3960, {n, 22}]
Accumulate[Accumulate[Accumulate[Range[22]^8]]]
CoefficientList[Series[(1 + 247 x + 4293 x^2 + 15619 x^3 + 15619 x^4 + 4293 x^5 + 247 x^6 + x^7)/(- 1 + x)^12, {x, 0, 22}], x]

Prog

(PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(3+2*n)*(1+36*n-69*n^2+45*n^4+18*n^5+2*n^6)/3960 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A000542, A001016, A022524, A253636.

Sequence in context: A038480 A309763 A344193 * A322882 A256594 A229433

Adjacent sequences: A254639 A254640 A254641 * A254643 A254644 A254645

Keyword

nonn,easy

Author

Luciano Ancora, Feb 05 2015

Status

approved