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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

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 (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. A000540, A001014, A022522, A101093, A101099.

Cf. A254644 (fourth partial sums of fifth powers), A254646 (fourth partial sums of seventh powers).

Sequence in context: A250144 A250338 A223374 * A281049 A264316 A251939

Adjacent sequences: A254642 A254643 A254644 * A254646 A254647 A254648

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 05 2015

STATUS

approved

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Last modified November 30 06:45 EST 2022. Contains 358431 sequences. (Running on oeis4.)