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
Keywords
Examples
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)
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.
Programs
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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 *) -
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 */
Formula
a(n) = A294775(n-1,2). - Alois P. Heinz, Nov 08 2017
Extensions
Extended by Jonathan Vos Post, Aug 30 2011
Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012.
Comments