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%I #12 Jan 10 2014 10:36:37
%S 0,0,0,1,1,2,4,7,13,24,43,78,140,251,452,812,1457,2617,4697,8428,
%T 15126,27142,48700,87384,156787,281307,504723,905562,1624731,2915039,
%U 5230040,9383505,16835453,30205347,54192931,97230224,174445475,312981054,561534340,1007475560
%N Restricted partitions.
%C Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=4 and p(k) <= 2*p(k+1), see example. [_Joerg Arndt_, Dec 18 2012]
%D Minc, H.; A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
%H Shimon Even and Abraham Lempel, <a href="http://dx.doi.org/10.1016/S0019-9958(72)90149-0">Generation and enumeration of all solutions of the characteristic sum condition</a>, Information and Control 21 (1972), 476-482.
%e From _Joerg Arndt_, Dec 18 2012: (Start)
%e There are a(9)=13 compositions 9=p(1)+p(2)+...+p(m) with p(1)=4 and p(k) <= 2*p(k+1):
%e [ 1] [ 3 1 1 1 1 1 ]
%e [ 2] [ 3 1 1 1 2 ]
%e [ 3] [ 3 1 1 2 1 ]
%e [ 4] [ 3 1 2 1 1 ]
%e [ 5] [ 3 1 2 2 ]
%e [ 6] [ 3 2 1 1 1 ]
%e [ 7] [ 3 2 1 2 ]
%e [ 8] [ 3 2 2 1 ]
%e [ 9] [ 3 2 3 ]
%e [10] [ 3 3 1 1 ]
%e [11] [ 3 3 2 ]
%e [12] [ 3 4 1 ]
%e [13] [ 3 5 ]
%e (End)
%p v := proc(c,d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i,d-c),i=1..2*c); fi; end; [ seq(v(4,n), n=1..50) ];
%t v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d-c], {i, 1, 2*c}]]]; Table[v[4, n], {n, 1, 40}] (* _Jean-François Alcover_, Jan 10 2014, translated from Maple *)
%Y Cf. A002572, A002573, A002574, A049285.
%K nonn,easy
%O 1,6
%A _N. J. A. Sloane_, Michael Somos
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