A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

1, 1, 1, 2, 4, 8, 16, 31, 61, 121, 240, 476, 944, 1872, 3712, 7362, 14601, 28958, 57432, 113904, 225904, 448034, 888583, 1762321, 3495200, 6932008, 13748208, 27266738, 54077957, 107252486, 212713209, 421872826, 836697836, 1659417786, 3291113315, 6527245245, 12945446241, 25674625681

Offset: 1

Author

Jonathan Vos Post, Aug 30 2011

Keywords

• nonn

Comments

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1), see example. - Joerg Arndt, Dec 18 2012

Row 4 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, and row 3 being A176503.

Examples

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(6+1)=16 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1):
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 1 3 ]
[ 5]  [ 1 1 2 1 1 ]
[ 6]  [ 1 1 2 2 ]
[ 7]  [ 1 1 3 1 ]
[ 8]  [ 1 1 4 ]
[ 9]  [ 1 2 1 1 1 ]
[10]  [ 1 2 1 2 ]
[11]  [ 1 2 2 1 ]
[12]  [ 1 2 3 ]
[13]  [ 1 3 1 1 ]
[14]  [ 1 3 2 ]
[15]  [ 1 4 1 ]
[16]  [ 1 5 ]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Crossrefs

Cf. A002572, A176485, A176503, A294775.

Programs

Mathematica

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n - j, k (r - j), k], {j, 0, Min[n, r]}]]];
a[n_] := b[4n - 3, 1, 5];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI

/* see A002572, set t=5 */

Formula

a(n) = A294775(n-1,4). - Alois P. Heinz, Nov 08 2017

Extensions

Terms beyond a(20)=113904 added by Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017