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A049286 Triangle of partitions v(d,c) defined in A002572.

%I A049286 #22 Jan 15 2024 12:02:18
%S A049286 1,1,1,2,1,1,3,2,1,1,5,4,2,1,1,9,7,4,2,1,1,16,12,7,4,2,1,1,28,22,13,7,
%T A049286 4,2,1,1,50,39,24,13,7,4,2,1,1,89,70,42,24,13,7,4,2,1,1,159,126,76,43,
%U A049286 24,13,7,4,2,1,1,285,225,137,78,43,24,13,7,4,2,1,1,510
%N A049286 Triangle of partitions v(d,c) defined in A002572.
%C A049286 Rows are the columns in the table at the end of the Minc reference, read top to bottom. - _Joerg Arndt_, Jan 15 2024
%H A049286 Shimon Even and 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 A049286 H. Minc, <a href="https://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 A049286 Triangle begins
%e A049286     1,
%e A049286     1,   1,
%e A049286     2,   1,   1,
%e A049286     3,   2,   1,  1,
%e A049286     5,   4,   2,  1,  1,
%e A049286     9,   7,   4,  2,  1,  1,
%e A049286    16,  12,   7,  4,  2,  1,  1,
%e A049286    28,  22,  13,  7,  4,  2,  1, 1,
%e A049286    50,  39,  24, 13,  7,  4,  2, 1, 1,
%e A049286    89,  70,  42, 24, 13,  7,  4, 2, 1, 1,
%e A049286   159, 126,  76, 43, 24, 13,  7, 4, 2, 1, 1,
%e A049286   285, 225, 137, 78, 43, 24, 13, 7, 4, 2, 1, 1,
%e A049286   ...
%e A049286 Rows read backward approach A002843. - _Joerg Arndt_, Jan 15 2024
%p A049286 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;
%t A049286 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[d, c], {c, 1, 13}, {d, 1, c}] // Flatten (* _Jean-François Alcover_, Dec 10 2012, after Maple *)
%Y A049286 Cf. A002572, A002573, A002574, A049284, A049285.
%Y A049286 See A047913 for another version.
%K A049286 nonn,tabl,nice,easy
%O A049286 1,4
%A A049286 _N. J. A. Sloane_, _Michael Somos_