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A254645 Fourth partial sums of sixth powers (A001014). 7
1, 68, 995, 7672, 40614, 166992, 571626, 1701480, 4534959, 11050468, 24997973, 53113424, 106959580, 205628736, 379603812, 676144944, 1166649837, 1956528420, 3198236503, 5108229896, 7988730530, 12255340240 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
G.f.: x*(1 + 57*x + 302*x^2 + 302*x^3 + 57*x^4 + x^5)/(1 - x)^11.
a(n) = n*(1 + n)*(2 + n)^2*(3 + n)*(4 + n)*(- 1 - 8*n + 14*n^2 + 8*n^3 + n^4)/5040.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^6.
EXAMPLE
First differences: 1, 63, 665, 3367, 11529, 31031, ... (A022522)
--------------------------------------------------------------------------
The sixth powers: 1, 64, 729, 4096, 15625, 46656, ... (A001014)
--------------------------------------------------------------------------
First partial sums: 1, 65, 794, 4890, 20515, 67171, ... (A000540)
Second partial sums: 1, 66, 860, 5750, 26265, 93436, ... (A101093)
Third partial sums: 1, 67, 927, 6677, 32942, 126378, ... (A101099)
Fourth partial sums: 1, 68, 995, 7672, 40614, 166992, ... (this sequence)
MAPLE
seq(binomial(n+4, 5)*(n+2)*((n^2+4*n-1)^2-2)/42, n=1..30); # G. C. Greubel, Aug 28 2019
MATHEMATICA
Table[n (1 + n) (2 + n)^2 (3 + n) (4 + n) (- 1 - 8 n + 14 n^2 + 8 n^3 + n^4)/5040, {n, 22}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[22]^6]]]] (* or *)
CoefficientList[Series[(- 1 - 57 x - 302 x^2 - 302 x^3 - 57 x^4 - x^5)/(- 1 + x)^11, {x, 0, 21}], x]
Nest[Accumulate, Range[30]^6, 4] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 68, 995, 7672, 40614, 166992, 571626, 1701480, 4534959, 11050468, 24997973}, 30] (* Harvey P. Dale, Dec 27 2015 *)
PROG
(PARI) vector(30, n, binomial(n+4, 5)*(n+2)*((n^2+4*n-1)^2-2)/42) \\ G. C. Greubel, Aug 28 2019
(Magma) [Binomial(n+4, 5)*(n+2)*((n^2+4*n-1)^2-2)/42: n in [1..30]]; // G. C. Greubel, Aug 28 2019
(Sage) [binomial(n+4, 5)*(n+2)*((n^2+4*n-1)^2-2)/42 for n in (1..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([1..30], n-> Binomial(n+4, 5)*(n+2)*((n^2+4*n-1)^2-2)/42); # G. C. Greubel, Aug 28 2019
CROSSREFS
Cf. A254644 (fourth partial sums of fifth powers), A254646 (fourth partial sums of seventh powers).
Sequence in context: A250338 A223374 A360656 * A281049 A264316 A251939
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 05 2015
STATUS
approved

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Last modified March 28 15:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)