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A254641 Third partial sums of seventh powers (A001015). 5
1, 131, 2577, 23723, 141694, 636426, 2331462, 7323954, 20396871, 51550213, 120271151, 262391493, 540659756, 1060489444, 1992739932, 3605846676, 6310148349, 10717864983, 17722868317, 28605158351, 45165823626, 69899222030, 106210179010, 158685165990 (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

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 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11.

a(n) = n*(1+n)*(2+n)*(3+n)*(6 -6*n -20*n^2 +15*n^3 +25*n^4 +9*n^5 +n^6)/720.

E.g.f.: x (720 +46440*x +262440*x^2 +425910*x^3 +285264*x^4 +92526*x^5 +15600*x^6 +1380*x^7 +60*x^8 +x^9)*exp(x)/6!. - G. C. Greubel, Aug 28 2019

MAPLE

seq(binomial(n+3, 4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30, n=1..30); # G. C. Greubel, Aug 28 2019

MATHEMATICA

Table[n(1+n)(2+n)(3+n)(6 -6n -20n^2 +15n^3 +25n^4 +9n^5 +n^6)/720, {n, 30}]

CoefficientList[Series[(1 +120x +1191x^2 +2416x^3 +1191x^4 +120x^5 + x^6)/(1-x)^11, {x, 0, 30}], x]

Nest[Accumulate, Range[30]^7, 3] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 131, 2577, 23723, 141694, 636426, 2331462, 7323954, 20396871, 51550213, 120271151}, 30] (* Harvey P. Dale, Jun 19 2018 *)

PROG

(PARI) Vec((1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11 + O(x^40)) \\ Andrew Howroyd, Nov 06 2018

(PARI) vector(30, n, binomial(n+3, 4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30) \\ G. C. Greubel, Aug 28 2019

(MAGMA) [Binomial(n+3, 4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30: n in [1..30]]; // G. C. Greubel, Aug 28 2019

(Sage) [binomial(n+3, 4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30 for n in (1..30)] # G. C. Greubel, Aug 28 2019

(GAP) List([1..30], n-> Binomial(n+3, 4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30); # G. C. Greubel, Aug 28 2019

CROSSREFS

Cf. A254640, A254642, A254643.

Sequence in context: A221154 A144247 A161197 * A322881 A104596 A091743

Adjacent sequences:  A254638 A254639 A254640 * A254642 A254643 A254644

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 05 2015

STATUS

approved

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Last modified August 6 08:46 EDT 2020. Contains 336228 sequences. (Running on oeis4.)