A254683 Fifth partial sums of sixth powers (A001014).

1, 69, 1064, 8736, 49350, 216342, 787968, 2489448, 7024407, 18074875, 43072848, 96186272, 203145852, 408774588, 788378400, 1464523344, 2631173181, 4587701601, 7785938104, 12894168000, 20882898530, 33138238770

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 + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^12.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(5 + 2*n)*(- 3 + 5*n + n^2)*(4 + 15*n + 3*n^2)/332640.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^6.

Example

First differences:   1, 63,  665, 3367, 11529, ...  (A022522)
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The sixth powers:    1, 64,  729, 4096, 15625, ...  (A001014)
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First partial sums:  1, 65,  794, 4890, 20515, ...  (A000540)
Second partial sums: 1, 66,  860, 5750, 26265, ...  (A101093)
Third partial sums:  1, 67,  927, 6677, 32942, ...  (A254640)
Fourth partial sums: 1, 68,  995, 7672, 40614, ...  (A254645)
Fifth partial sums:  1, 69, 1064, 8736, 49350, ...  (this sequence)

Mathematica

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (5 + 2*n) (- 3 + 5*n + n^2) (4 + 15 n + 3 n^2)/332640, {n, 22}] (* or *)
CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^12, {x, 0, 21}], x]

Crossrefs

Cf. A000540, A001014, A022522, A101093, A254640, A254645, A254681, A254682, A254684.

Sequence in context: A160834 A160833 A160831 * A095255 A253000 A295212

Adjacent sequences: A254680 A254681 A254682 * A254684 A254685 A254686

Keyword

nonn,easy

Author

Luciano Ancora, Feb 12 2015

Status

approved