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
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 05 2015
STATUS
approved