A276430 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).
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, 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, 17, 10, 4, 2, 1, 43, 49, 37, 24, 13, 6, 3, 1, 52, 63, 50, 31, 18, 10, 4, 2, 1
Offset: 0
Examples
T(6,1) = 3, counting [1,1,4], [1,2,3], [1,1,1,1,2]; T(6,2) = 2, counting [2,4], [1,1,2,2]; T(6,3) = 1, counting [2,2,2]; Triangle starts: 1; 1; 1,1; 2,1; 2,2,1; 3,3,1; ...
Links
- Alois P. Heinz, Rows n = 0..300, flattened
Programs
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Maple
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 # second Maple program: p2:= proc(n) p2(n):= is(n=2^ilog2(n)) end: b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)*`if`(p2(i), x, 1)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)): seq(T(n), n=0..30); # Alois P. Heinz, Sep 20 2016 -
Mathematica
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 *)
Formula
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.
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