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A002574 Restricted partitions.

%I A002574 M1070 N0404 #32 Oct 17 2023 05:26:00
%S A002574 0,0,1,1,2,4,7,13,24,42,76,137,245,441,792,1420,2550,4576,8209,14732,
%T A002574 26433,47424,85092,152670,273914,491453,881744,1581985,2838333,
%U A002574 5092398,9136528,16392311,29410243,52766343,94670652,169853138,304741614,546751437,980952673,1759973660
%N A002574 Restricted partitions.
%C A002574 Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1), see example. [_Joerg Arndt_, Dec 18 2012]
%D A002574 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002574 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002574 Vincenzo Librandi, <a href="/A002574/b002574.txt">Table of n, a(n) for n = 1..200</a>
%H A002574 Shimon Even and Abraham Lempel, <a href="https://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 A002574 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 A002574 From _Joerg Arndt_, Dec 18 2012: (Start)
%e A002574 There are a(8)=13 compositions 8=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1):
%e A002574 [ 1]  [ 3 1 1 1 1 1 ]
%e A002574 [ 2]  [ 3 1 1 1 2 ]
%e A002574 [ 3]  [ 3 1 1 2 1 ]
%e A002574 [ 4]  [ 3 1 2 1 1 ]
%e A002574 [ 5]  [ 3 1 2 2 ]
%e A002574 [ 6]  [ 3 2 1 1 1 ]
%e A002574 [ 7]  [ 3 2 1 2 ]
%e A002574 [ 8]  [ 3 2 2 1 ]
%e A002574 [ 9]  [ 3 2 3 ]
%e A002574 [10]  [ 3 3 1 1 ]
%e A002574 [11]  [ 3 3 2 ]
%e A002574 [12]  [ 3 4 1 ]
%e A002574 [13]  [ 3 5 ]
%e A002574 (End)
%p A002574 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(3,n), n=1..50) ];
%t A002574 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[3, n]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 05 2013, after Maple *)
%Y A002574 Cf. A002572, A002573, A049284, A049285, A047913.
%K A002574 nonn,easy
%O A002574 1,5
%A A002574 _N. J. A. Sloane_
%E A002574 More terms from _Michael Somos_