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%I #42 May 26 2023 10:11:47
%S 0,0,1,1,2,3,5,7,10,14,20,27,37,49,66,86,113,147,190,243,311,394,499,
%T 627,786,980,1220,1510,1865,2294,2816,3443,4202,5110,6203,7507,9067,
%U 10923,13135,15755,18865,22540,26885,32001,38032,45112,53430,63171
%N Number of Garden of Eden partitions of n in Bulgarian Solitaire.
%C a(n) gives the number of times n occurs in A225794. - _Antti Karttunen_, Jul 27 2013
%H Seiichi Manyama, <a href="/A123975/b123975.txt">Table of n, a(n) for n = 1..10000</a>
%H Brian Hopkins, Michael A. Jones, <a href="https://doi.org/10.37236/1106">Shift-Induced Dynamical Systems on Partitions and Compositions</a>, Electron. J. Combin. 13 (2006), Research Paper 80.
%H Brian Hopkins and James A. Sellers, <a href="https://www.emis.de/journals/INTEGERS/papers/a19int2005/a19int2005.pdf">Exact enumeration of Garden of Eden partitions</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory 7(2) (2007), #A19.
%F a(n) = A064173(n) - A101198(n).
%F a(n) = Sum_{j>=1} (-1)^(j+1)*p(n-b(j)) where b(j) = 3*j*(j+1)/2 (A045943) and p(n) is the number of partitions of n (see A000041). See Hopkins & Sellers. - _Michel Marcus_, Sep 26 2018
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 19*Pi/144) / sqrt(n/6)). - _Vaclav Kotesovec_, May 26 2023
%p p:=product(1/(1-q^i), i=1..200)*sum((-1)^(r-1)*q^((3*r^2+3*r)/2), r=1..200):s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s, q, j)) od: # _James A. Sellers_, Nov 30 2006
%o (PARI) my(N=50, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k+1)/2)))) \\ _Seiichi Manyama_, May 21 2023
%Y Cf. A000041, A045943, A064173, A101198, A237831.
%Y Cf. A227753, A225794, A226062.
%K easy,nonn
%O 1,5
%A _Vladeta Jovovic_, Nov 23 2006
%E More terms from _James A. Sellers_, Nov 30 2006
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