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%I #23 Mar 17 2024 20:00:25
%S 1,1,1430,399675,22377814,508665365,6530486977,56519001176,
%T 366504924502,1906401762732,8333338333334,31645829208856,
%U 106993223294977,328114730182533,926000621503254,2432743920878907,6004799637378390,14031485751786081,31234447604616769
%N Number of partitions of n^7 into at most three parts.
%H Colin Barker, <a href="/A274253/b274253.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (13, -77, 274, -650, 1078, -1274, 1066, -572, 0, 572, -1066, 1274, -1078, 650, -274, 77, -13, 1).
%F Coefficient of x^(n^7) in 1/((1-x)*(1-x^2)*(1-x^3)).
%F G.f.: (1 -12*x +1494*x^2 +380888*x^3 +17292525*x^4 +248136510*x^5 +1532347656*x^6 +4916629962*x^7 +9347647209*x^8 +11464268960*x^9 +9347652702*x^10 +4916635404*x^11 +1532337619*x^12 +248138478*x^13 +17294340*x^14 +380562*x^15 +1302*x^16) / ((1 -x)^15*(1 +x)*(1 +x +x^2)).
%F a(n) = A001399(n^7) = round((n^7+3)^2/12). - _Alois P. Heinz_, Jun 16 2016
%t CoefficientList[Series[(1-12x+1494x^2+380888x^3+17292525x^4+248136510x^5+1532347656x^6+ 4916629962x^7+ 9347647209x^8+11464268960x^9+9347652702x^10+ 4916635404x^11+ 1532337619x^12+ 248138478x^13+17294340x^14+380562x^15+1302x^16)/((1-x)^15(1+x)(1+x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{13,-77,274,-650,1078,-1274,1066,-572,0,572,-1066,1274,-1078,650,-274,77,-13,1},{1,1,1430,399675,22377814,508665365,6530486977,56519001176,366504924502,1906401762732,8333338333334,31645829208856,106993223294977,328114730182533,926000621503254,2432743920878907,6004799637378390,14031485751786081},30] (* _Harvey P. Dale_, Dec 09 2022 *)
%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^7))
%Y A subsequence of A001399.
%Y Cf. A274250 (n^2), A274251 (n^3), A274252 (n^5), A274254 (n^11).
%K nonn,easy
%O 0,3
%A _Colin Barker_, Jun 16 2016
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