A049284 Restricted partitions.

0, 0, 0, 1, 1, 2, 4, 7, 13, 24, 43, 78, 140, 251, 452, 812, 1457, 2617, 4697, 8428, 15126, 27142, 48700, 87384, 156787, 281307, 504723, 905562, 1624731, 2915039, 5230040, 9383505, 16835453, 30205347, 54192931, 97230224, 174445475, 312981054, 561534340, 1007475560

Offset: 1

Author

N. J. A. Sloane, Michael Somos

Keywords

• nonn
• easy

Comments

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]

Examples

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(9)=13 compositions 9=p(1)+p(2)+...+p(m) with p(1)=4 and p(k) <= 2*p(k+1):
[ 1]  [ 3 1 1 1 1 1 ]
[ 2]  [ 3 1 1 1 2 ]
[ 3]  [ 3 1 1 2 1 ]
[ 4]  [ 3 1 2 1 1 ]
[ 5]  [ 3 1 2 2 ]
[ 6]  [ 3 2 1 1 1 ]
[ 7]  [ 3 2 1 2 ]
[ 8]  [ 3 2 2 1 ]
[ 9]  [ 3 2 3 ]
[10]  [ 3 3 1 1 ]
[11]  [ 3 3 2 ]
[12]  [ 3 4 1 ]
[13]  [ 3 5 ]
(End)
		

References

• 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.

Links

• Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.

Crossrefs

Cf. A002572, A002573, A002574, A049285.

Programs

Maple

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) ];

Mathematica

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 *)