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%I #16 Dec 21 2016 13:48:16
%S 1,1,1,1,2,1,2,2,1,3,3,1,5,3,2,1,6,5,3,1,7,8,4,2,1,10,10,6,3,1,13,13,
%T 9,4,2,1,16,18,12,6,3,1,22,22,16,10,4,2,1,27,29,22,13,6,3,1,33,40,28,
%U 17,10,4,2,1,43,49,37,24,13,6,3,1,52,63,50,31,18,10,4,2,1
%N Triangle read by rows: T(n,k) is the number of partitions of n having k parts that are powers of 2 with positive exponent (n>=0).
%C Sum of entries in row n is A000041(n) (the partition numbers).
%C T(n,0) = A276431(n).
%C Sum(k*T(n,k), k>=0) = A073119(n).
%H Alois P. Heinz, <a href="/A276430/b276430.txt">Rows n = 0..300, flattened</a>
%F G.f.: G(t,x) = Product_{i>=1} (1-x^{h(i)})/((1-x^i)*(1-t*x^{h(i)})), where h(i) = 2^i.
%e T(6,1) = 3, counting [1,1,4], [1,2,3], [1,1,1,1,2];
%e T(6,2) = 2, counting [2,4], [1,1,2,2];
%e T(6,3) = 1, counting [2,2,2];
%e Triangle starts:
%e 1;
%e 1;
%e 1,1;
%e 2,1;
%e 2,2,1;
%e 3,3,1;
%e ...
%p h := proc (i) options operator, arrow: 2^i 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, 25)): for n from 0 to 20 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p # second Maple program:
%p p2:= proc(n) p2(n):= is(n=2^ilog2(n)) end:
%p b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1,
%p b(n, i-1)+`if`(i>n, 0, b(n-i, i)*`if`(p2(i), x, 1))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..30); # _Alois P. Heinz_, Sep 20 2016
%t p2[n_] := n == 2^Floor[Log[2, n]]; b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1, b[n, i-1] + If[i>n, 0, b[n-i, i]*If[p2[i], x, 1]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%Y Cf. A000041, A073119, A276431.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Sep 19 2016
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