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A274323 Number of partitions of n^4 into at most two parts.

%I A274323 #19 Mar 17 2024 23:38:02
%S A274323 1,1,9,41,129,313,649,1201,2049,3281,5001,7321,10369,14281,19209,
%T A274323 25313,32769,41761,52489,65161,80001,97241,117129,139921,165889,
%U A274323 195313,228489,265721,307329,353641,405001,461761,524289,592961,668169,750313,839809,937081
%N A274323 Number of partitions of n^4 into at most two parts.
%C A274323 Coefficient of x^(n^4) in 1/((1-x)*(1-x^2)).
%H A274323 Colin Barker, <a href="/A274323/b274323.txt">Table of n, a(n) for n = 0..1000</a>
%H A274323 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A274323 G.f.: (1 - 3*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1-x)^5*(1+x)).
%F A274323 a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n > 5.
%F A274323 a(n) = (3 + (-1)^n + 2*n^4)/4.
%F A274323 a(n) = A008619(n^4).
%F A274323 a(n) = 1 + floor(n^4/2). - _Alois P. Heinz_, Oct 13 2016
%F A274323 E.g.f.: ((2 + x + 7*x^2 + 6*x^3 + x^4)*cosh(x) + (1 + x + 7*x^2 + 6*x^3 + x^4)*sinh(x))/2. - _Stefano Spezia_, Mar 17 2024
%o A274323 (PARI) a(n) = (3+(-1)^n+2*n^4)/4
%o A274323 (PARI)
%o A274323 b(n) = (3+(-1)^n+2*n)/4 \\ the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2))
%o A274323 vector(50, n, n--; b(n^4))
%Y A274323 Cf. A099392 (n^2), A274324 (n^3), A274325 (n^5).
%Y A274323 Cf. A008619.
%K A274323 nonn,easy
%O A274323 0,3
%A A274323 _Colin Barker_, Oct 13 2016