A089052 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of n into exactly k powers of 2.
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1
Offset: 0
Examples
1 0 1 0 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 1 2 1 1 1 0 0 0 1 2 1 1 1 0 1 1 1 2 2 1 1 1 0 0 1 1 1 2 2 1 1 1 0 0 1 2 2 2 2 2 1 1 1 0 0 0 1 2 2 2 2 2 1 1 1
References
- J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From N. J. A. Sloane, Feb 03 2013
Links
- Alois P. Heinz, Rows n = 0..200, flattened
- J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From _N. J. A. Sloane_, Feb 03 2013
Crossrefs
Programs
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Maple
A089052 := proc(n, k) option remember; if k > n then return(0); end if; if k= 0 then if n=0 then return(1) else return(0); end if; end if; if n mod 2 = 1 then return procname(n-1, k-1); end if; procname(n-1, k-1)+procname(n/2, k); end proc:
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Mathematica
t[n_, k_] := t[n, k] = Which[k > n, 0, k == 0, If[n == 0, 1, 0], Mod[n, 2] == 1, t[n-1, k-1], True, t[n-1, k-1] + t[n/2, k]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Maple *)
Formula
T(2m, k) = T(m, k)+T(2m-1, k-1); T(2m+1, k) = T(2m, k-1).
G.f.: 1/Product_{k>=0} (1-y*x^(2^k)). - Vladeta Jovovic, Dec 03 2003