A123975 Number of Garden of Eden partitions of n in Bulgarian Solitaire.
0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 147, 190, 243, 311, 394, 499, 627, 786, 980, 1220, 1510, 1865, 2294, 2816, 3443, 4202, 5110, 6203, 7507, 9067, 10923, 13135, 15755, 18865, 22540, 26885, 32001, 38032, 45112, 53430, 63171
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- Brian Hopkins, Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80.
- Brian Hopkins and James A. Sellers, Exact enumeration of Garden of Eden partitions, INTEGERS: Electronic Journal of Combinatorial Number Theory 7(2) (2007), #A19.
Programs
-
Maple
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 Sellers, Nov 30 2006
-
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
Formula
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
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
Extensions
More terms from James Sellers, Nov 30 2006
Comments