%I #13 Jun 10 2019 02:12:04

%S 0,0,1,2,3,4,6,8,10,14,18,22,29,36,44,56,68,82,101,122,146,176,210,

%T 248,296,350,410,484,566,660,772,896,1038,1204,1391,1602,1846,2120,

%U 2428,2784,3182,3628,4138,4708,5347,6072,6880,7784,8804,9940,11208,12630

%N Number of plasma partitions of 2n-1.

%H Seiichi Manyama, <a href="/A095913/b095913.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: sum(i>=1, x^(i+2)/prod(j=1..i, 1-x^(2*j-1))) . - _Michael Somos_, Aug 18 2006

%F G.f.: x^2*(1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 18 2013

%F a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - _Vaclav Kotesovec_, Jun 10 2019

%e A plasma partition is a partition of n into 1 distinct odd part and an even number of odd parts and at least 2 parts of 1, so looking like plasma.

%e E.g. a(7) counts the plasma partitions of 13, has 11+1+1 = 9+1+1 = 7+1+1+1+1 = 5+1+1+1+1+1+1 = 5+3+3+1+1 = 3+1+1+1+1+1+1+1+1, so a(7)=6.

%e Graphically, these are;

%e .....*..........*........*......*.....*....*

%e ***********.....*........*......*....***...*

%e .....*......*********....*......*...*****..*

%e ................*.....*******...*....***...*

%e ................*........*....*****...*....*

%e .........................*......*.........***

%e .........................*......*..........*

%e ................................*..........*

%e ................................*..........*

%e ...........................................*

%e ...........................................*

%o (PARI) {a(n)=local(A); if(n<3, 0, n-=2; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* _Michael Somos_, Aug 18 2006 */

%Y a(n)=A053253(n-3).

%K nonn

%O 1,4

%A _Jon Perry_, Jul 13 2004