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%I A002573 M1062 N0399 #32 Dec 20 2021 20:22:22
%S A002573 0,1,1,2,4,7,12,22,39,70,126,225,404,725,1299,2331,4182,7501,13458,
%T A002573 24145,43316,77715,139430,250152,448808,805222,1444677,2591958,
%U A002573 4650342,8343380,14969239,26856992,48185362,86451602,155106844,278284440,499283177,895787396,1607174300,2883507098
%N A002573 Restricted partitions.
%C A002573 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 A002573 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002573 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002573 Vincenzo Librandi, <a href="/A002573/b002573.txt">Table of n, a(n) for n = 1..500</a>
%H A002573 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 A002573 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 A002573 From _Joerg Arndt_, Dec 18 2012: (Start)
%e A002573 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 A002573 [ 1]  [ 2 1 1 1 1 1 1 ]
%e A002573 [ 2]  [ 2 1 1 1 1 2 ]
%e A002573 [ 3]  [ 2 1 1 1 2 1 ]
%e A002573 [ 4]  [ 2 1 1 2 1 1 ]
%e A002573 [ 5]  [ 2 1 1 2 2 ]
%e A002573 [ 6]  [ 2 1 2 1 1 1 ]
%e A002573 [ 7]  [ 2 1 2 1 2 ]
%e A002573 [ 8]  [ 2 1 2 2 1 ]
%e A002573 [ 9]  [ 2 1 2 3 ]
%e A002573 [10]  [ 2 2 1 1 1 1 ]
%e A002573 [11]  [ 2 2 1 1 2 ]
%e A002573 [12]  [ 2 2 1 2 1 ]
%e A002573 [13]  [ 2 2 2 1 1 ]
%e A002573 [14]  [ 2 2 2 2 ]
%e A002573 [15]  [ 2 2 3 1 ]
%e A002573 [16]  [ 2 2 4 ]
%e A002573 [17]  [ 2 3 1 1 1 ]
%e A002573 [18]  [ 2 3 1 2 ]
%e A002573 [19]  [ 2 3 2 1 ]
%e A002573 [20]  [ 2 3 3 ]
%e A002573 [21]  [ 2 4 1 1 ]
%e A002573 [22]  [ 2 4 2 ]
%e A002573 (End)
%p A002573 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 A002573 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 A002573 Cf. A002572, A047913, A002574, A049284, A049285.
%Y A002573 A column of the triangle in A176431.
%K A002573 nonn,easy
%O A002573 1,4
%A A002573 _N. J. A. Sloane_

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