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%I #11 Nov 13 2015 16:18:37
%S 1,0,1,1,0,1,0,2,0,1,2,0,2,0,1,1,3,0,2,0,1,3,1,4,0,2,0,1,1,6,1,4,0,2,
%T 0,1,5,2,7,1,4,0,2,0,1,3,9,2,8,1,4,0,2,0,1,8,4,12,2,8,1,4,0,2,0,1,5,
%U 15,5,13,2,8,1,4,0,2,0,1,12,9,19,5,14,2,8,1,4,0,2,0,1
%N Triangle read by rows: T(n,k) is the number of partitions of n having k Mersenne number parts (0<=k<=n).
%C The Mersenne numbers are of the form 2^n - 1 (n >= 0), i.e., 0, 1, 3, 7, 15, 31, ....; A000225.
%C Sum of entries in row n = A000041(n) = number of partitions of n.
%C T(n,0) = A078657(n).
%C Sum_{k=0..n} k*T(n,k) = A264395(n) = total number of Mersenne number parts in all partitions of n.
%F G.f.: G(t,x) = Product_{i>0} (1-x^(h(i)))/((1-x^i)*(1-t*x^(h(i)))), where h(i) = 2^i - 1.
%e T(7,3) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,1,1,1] (the partitions of 7 that have 3 Mersenne number parts).
%e Triangle starts:
%e 1;
%e 0,1;
%e 1,0,1;
%e 0,2,0,1;
%e 2,0,2,0,1;
%e 1,3,0,2,0,1;
%p h := proc (i) options operator, arrow: 2^i-1 end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 30): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
%Y Cf. A000041, A000225, A078657, A264395.
%K nonn,tabl
%O 0,8
%A _Emeric Deutsch_, Nov 13 2015
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