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A274324 Number of partitions of n^3 into at most two parts.

%I A274324 #21 Sep 28 2022 17:27:32
%S A274324 1,1,5,14,33,63,109,172,257,365,501,666,865,1099,1373,1688,2049,2457,
%T A274324 2917,3430,4001,4631,5325,6084,6913,7813,8789,9842,10977,12195,13501,
%U A274324 14896,16385,17969,19653,21438,23329,25327,27437,29660,32001,34461,37045,39754
%N A274324 Number of partitions of n^3 into at most two parts.
%H A274324 Colin Barker, <a href="/A274324/b274324.txt">Table of n, a(n) for n = 0..1000</a>
%H A274324 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A274324 Coefficient of x^(n^3) in 1/((1-x)*(1-x^2)).
%F A274324 a(n) = A008619(n^3).
%F A274324 a(n) = (3+(-1)^n+2*n^3)/4.
%F A274324 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.
%F A274324 G.f.: (1-2*x+4*x^2+3*x^3) / ((1-x)^4*(1+x)).
%F A274324 From _Stefano Spezia_, Sep 28 2022: (Start)
%F A274324 a(n) = A050492((n+1)/2) for n odd.
%F A274324 E.g.f.: ((2 + x + 3*x^2 + x^3)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. (End)
%p A274324 A274324:=n->(3+(-1)^n+2*n^3)/4: seq(A274324(n), n=0..50); # _Wesley Ivan Hurt_, Jun 25 2016
%t A274324 Table[(3+(-1)^n+2*n^3)/4, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 25 2016 *)
%o A274324 (PARI)
%o A274324 \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)).
%o A274324 b(n) = (3+(-1)^n+2*n)/4
%o A274324 vector(50, n, n--; b(n^3))
%o A274324 (Magma) [(3+(-1)^n+2*n^3)/4 : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 25 2016
%Y A274324 A subsequence of A008619.
%Y A274324 Cf. A099392 (n^2), A274325 (n^5).
%Y A274324 Cf. also A050492.
%K A274324 nonn,easy
%O A274324 0,3
%A A274324 _Colin Barker_, Jun 18 2016