A254645 - OEIS

A254645

Fourth partial sums of sixth powers (A001014).

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: A250338 A223374 A360656 * A281049 A264316 A251939

Adjacent sequences: A254642 A254643 A254644 * A254646 A254647 A254648

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 05 2015

STATUS

approved