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%I #22 Mar 17 2024 23:38:07
%S 1,1,17,122,513,1563,3889,8404,16385,29525,50001,80526,124417,185647,
%T 268913,379688,524289,709929,944785,1238050,1600001,2042051,2576817,
%U 3218172,3981313,4882813,5940689,7174454,8605185,10255575,12150001,14314576,16777217
%N Number of partitions of n^5 into at most two parts.
%H Colin Barker, <a href="/A274325/b274325.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,5,5,-9,5,-1).
%F Coefficient of x^(n^5) in 1/((1-x)*(1-x^2)).
%F a(n) = A008619(n^5).
%F a(n) = (3 + (-1)^n + 2*n^5)/4.
%F a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n > 6.
%F G.f.: (1 - 4*x + 21*x^2 + 41*x^3 + 46*x^4 + 15*x^5) / ((1-x)^6*(1+x)).
%F E.g.f.: ((2 + x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)*cosh(x) + (1 + x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)*sinh(x))/2. - _Stefano Spezia_, Mar 17 2024
%p A274325:=n->(3+(-1)^n+2*n^5)/4: seq(A274325(n), n=0..50); # _Wesley Ivan Hurt_, Jun 25 2016
%t Table[(3+(-1)^n+2*n^5)/4, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 25 2016 *)
%o (PARI)
%o \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)).
%o b(n) = (3+(-1)^n+2*n)/4
%o vector(50, n, n--; b(n^5))
%o (Magma) [(3+(-1)^n+2*n^5)/4 : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 25 2016
%Y A subsequence of A008619.
%Y Cf. A099392 (n^2), A274324 (n^3).
%K nonn,easy
%O 0,3
%A _Colin Barker_, Jun 18 2016
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