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

1,6

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]

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

Table of n, a(n) for n=1..40.

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

Example

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)

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 *)

Crossrefs

Cf. A002572, A002573, A002574, A049285.

Sequence in context: A069765 A090427 A006745 * A049285 A002843 A128742

Adjacent sequences: A049281 A049282 A049283 * A049285 A049286 A049287

Keyword

nonn,easy

Author

N. J. A. Sloane, Michael Somos

Status

approved