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%I #18 Mar 17 2024 20:04:28
%S 1,1,102,5043,87894,815365,5042737,23548008,89494870,290594892,
%T 833383334,2161532576,5159904769,11488393301,24104823494,48054578907,
%U 91626493270,168000201633,297539880337,510923426200,853334933334,1389992123568,2213329476102,3452212485976
%N Number of partitions of n^5 into at most three parts.
%H Colin Barker, <a href="/A274252/b274252.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (9, -35, 76, -99, 77, -33, 0, 33, -77, 99, -76, 35, -9, 1).
%F Coefficient of x^(n^5) in 1/((1-x)*(1-x^2)*(1-x^3)).
%F G.f.: (1 -8*x +128*x^2 +4084*x^3 +46100*x^4 +193094*x^5 +407528*x^6 +512642*x^7 +407381*x^8 +193090*x^9 +46120*x^10 +4170*x^11 +70*x^12) / ((1 -x)^11*(1 +x)*(1 +x +x^2)).
%F a(n) = A001399(n^5) = round((n^5+3)^2/12). - _Alois P. Heinz_, Jun 16 2016
%t CoefficientList[Series[(1-8x+128x^2+4084x^3+46100x^4+193094x^5+407528x^6+512642x^7+407381x^8+193090x^9+46120x^10+4170x^11+70x^12)/((1-x)^11(1+x)(1+x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{9,-35,76,-99,77,-33,0,33,-77,99,-76,35,-9,1},{1,1,102,5043,87894,815365,5042737,23548008,89494870,290594892,833383334,2161532576,5159904769,11488393301},30] (* _Harvey P. Dale_, May 15 2023 *)
%o (PARI)
%o \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)).
%o b(n) = round(real((47+9*(-1)^n + 8*exp(-2/3*I*n*Pi) + 8*exp((2*I*n*Pi)/3) + 36*n+6*n^2)/72))
%o vector(50, n, n--; b(n^5))
%Y A subsequence of A001399.
%Y Cf. A274250 (n^2), A274251 (n^3), A274253 (n^7), A274254 (n^11).
%K nonn,easy
%O 0,3
%A _Colin Barker_, Jun 16 2016