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

Author

Luciano Ancora, Feb 12 2015

Keywords

• nonn
• easy

Examples

			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)
		

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).

Crossrefs

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

Programs

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]

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.