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Restricted partitions.
(Formerly M1062 N0399)
9

%I M1062 N0399 #32 Dec 20 2021 20:22:22

%S 0,1,1,2,4,7,12,22,39,70,126,225,404,725,1299,2331,4182,7501,13458,

%T 24145,43316,77715,139430,250152,448808,805222,1444677,2591958,

%U 4650342,8343380,14969239,26856992,48185362,86451602,155106844,278284440,499283177,895787396,1607174300,2883507098

%N Restricted partitions.

%C Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1), see example. [_Joerg Arndt_, Dec 18 2012]

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002573/b002573.txt">Table of n, a(n) for n = 1..500</a>

%H Shimon Even & 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.

%H H. Minc, <a href="http://dx.doi.org/10.1017/S0013091500021945">A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid</a>, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

%e From _Joerg Arndt_, Dec 18 2012: (Start)

%e There are a(8)=22 compositions 8=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1):

%e [ 1] [ 2 1 1 1 1 1 1 ]

%e [ 2] [ 2 1 1 1 1 2 ]

%e [ 3] [ 2 1 1 1 2 1 ]

%e [ 4] [ 2 1 1 2 1 1 ]

%e [ 5] [ 2 1 1 2 2 ]

%e [ 6] [ 2 1 2 1 1 1 ]

%e [ 7] [ 2 1 2 1 2 ]

%e [ 8] [ 2 1 2 2 1 ]

%e [ 9] [ 2 1 2 3 ]

%e [10] [ 2 2 1 1 1 1 ]

%e [11] [ 2 2 1 1 2 ]

%e [12] [ 2 2 1 2 1 ]

%e [13] [ 2 2 2 1 1 ]

%e [14] [ 2 2 2 2 ]

%e [15] [ 2 2 3 1 ]

%e [16] [ 2 2 4 ]

%e [17] [ 2 3 1 1 1 ]

%e [18] [ 2 3 1 2 ]

%e [19] [ 2 3 2 1 ]

%e [20] [ 2 3 3 ]

%e [21] [ 2 4 1 1 ]

%e [22] [ 2 4 2 ]

%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(2,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}]]]; a[n_] := v[2, n]; Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Jan 30 2012, after Maple *)

%Y Cf. A002572, A047913, A002574, A049284, A049285.

%Y A column of the triangle in A176431.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_