A176485 First column of triangle in A176452.

1, 1, 1, 2, 4, 7, 13, 25, 48, 92, 176, 338, 649, 1246, 2392, 4594, 8823, 16945, 32545, 62509, 120060, 230598, 442910, 850701, 1633948, 3138339, 6027842, 11577747, 22237515, 42711863, 82037200, 157569867, 302646401, 581296715, 1116503866, 2144482948, 4118935248, 7911290530

Offset

1,4

Comments

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

Row 2 of Table 1 of Elsholtz, row 1 being A002572. - Jonathan Vos Post, Aug 30 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.

Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Formula

a(n) = A294775(n-1,2). - Alois P. Heinz, Nov 08 2017

Example

From Joerg Arndt, Dec 18 2012: (Start)
There are a(7+1)=25 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*p(k+1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 1 2 ]
[ 3]  [ 1 1 1 1 2 1 ]
[ 4]  [ 1 1 1 1 3 ]
[ 5]  [ 1 1 1 2 1 1 ]
[ 6]  [ 1 1 1 2 2 ]
[ 7]  [ 1 1 1 3 1 ]
[ 8]  [ 1 1 2 1 1 1 ]
[ 9]  [ 1 1 2 1 2 ]
[10]  [ 1 1 2 2 1 ]
[11]  [ 1 1 2 3 ]
[12]  [ 1 1 3 1 1 ]
[13]  [ 1 1 3 2 ]
[14]  [ 1 2 1 1 1 1 ]
[15]  [ 1 2 1 1 2 ]
[16]  [ 1 2 1 2 1 ]
[17]  [ 1 2 1 3 ]
[18]  [ 1 2 2 1 1 ]
[19]  [ 1 2 2 2 ]
[20]  [ 1 2 3 1 ]
[21]  [ 1 2 4 ]
[22]  [ 1 3 1 1 1 ]
[23]  [ 1 3 1 2 ]
[24]  [ 1 3 2 1 ]
[25]  [ 1 3 3 ]
(End)

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[2n-1, 1, 3];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

Prog

(PARI)
/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=3;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
L=2 + 2*ceil( log(N) / log(t) );
f(k) = (1-t^k)/(1-t);
la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf = nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */

Crossrefs

Cf. A176452, A002572, A176503, A294775.

Sequence in context: A018083 A108361 A082423 * A119266 A102026 A103204

Adjacent sequences: A176482 A176483 A176484 * A176486 A176487 A176488

Keyword

nonn

Author

N. J. A. Sloane, Dec 07 2010

Extensions

Extended by Jonathan Vos Post, Aug 30 2011

Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012.

Status

approved