%I #41 Mar 21 2019 18:01:29

%S 1,1,1,2,4,7,13,25,48,92,176,338,649,1246,2392,4594,8823,16945,32545,

%T 62509,120060,230598,442910,850701,1633948,3138339,6027842,11577747,

%U 22237515,42711863,82037200,157569867,302646401,581296715,1116503866,2144482948,4118935248,7911290530

%N First column of triangle in A176452.

%C 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]

%C Row 2 of Table 1 of Elsholtz, row 1 being A002572. - _Jonathan Vos Post_, Aug 30 2011

%H Alois P. Heinz, <a href="/A176485/b176485.txt">Table of n, a(n) for n = 1..2000</a>

%H Christian Elsholtz, Clemens Heuberger, Daniel Krenn, <a href="https://arxiv.org/abs/1901.11343">Algorithmic counting of nonequivalent compact Huffman codes</a>, arXiv:1901.11343 [math.CO], 2019.

%H Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="https://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

%F a(n) = A294775(n-1,2). - _Alois P. Heinz_, Nov 08 2017

%e From _Joerg Arndt_, Dec 18 2012: (Start)

%e 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):

%e [ 1] [ 1 1 1 1 1 1 1 ]

%e [ 2] [ 1 1 1 1 1 2 ]

%e [ 3] [ 1 1 1 1 2 1 ]

%e [ 4] [ 1 1 1 1 3 ]

%e [ 5] [ 1 1 1 2 1 1 ]

%e [ 6] [ 1 1 1 2 2 ]

%e [ 7] [ 1 1 1 3 1 ]

%e [ 8] [ 1 1 2 1 1 1 ]

%e [ 9] [ 1 1 2 1 2 ]

%e [10] [ 1 1 2 2 1 ]

%e [11] [ 1 1 2 3 ]

%e [12] [ 1 1 3 1 1 ]

%e [13] [ 1 1 3 2 ]

%e [14] [ 1 2 1 1 1 1 ]

%e [15] [ 1 2 1 1 2 ]

%e [16] [ 1 2 1 2 1 ]

%e [17] [ 1 2 1 3 ]

%e [18] [ 1 2 2 1 1 ]

%e [19] [ 1 2 2 2 ]

%e [20] [ 1 2 3 1 ]

%e [21] [ 1 2 4 ]

%e [22] [ 1 3 1 1 1 ]

%e [23] [ 1 3 1 2 ]

%e [24] [ 1 3 2 1 ]

%e [25] [ 1 3 3 ]

%e (End)

%t 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]}]]];

%t a[n_] := b[2n-1, 1, 3];

%t Array[a, 40] (* _Jean-François Alcover_, Jul 21 2018, after _Alois P. Heinz_ *)

%o (PARI)

%o /* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */

%o N=66; q='q+O('q^N);

%o t=3; /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503 */

%o L=2 + 2*ceil( log(N) / log(t) );

%o f(k) = (1-t^k)/(1-t);

%o la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );

%o nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );

%o dn=sum(j=0, L, (-1)^j * la(j) );

%o gf = nm / dn;

%o Vec( gf )

%o /* _Joerg Arndt_, Dec 27 2012 */

%Y Cf. A176452, A002572, A176503, A294775.

%K nonn

%O 1,4

%A _N. J. A. Sloane_, Dec 07 2010

%E Extended by _Jonathan Vos Post_, Aug 30 2011

%E Added terms beyond a(20)=62509, _Joerg Arndt_, Dec 18 2012.