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%I #35 Feb 15 2020 23:54:54
%S 1,1,1,2,4,8,16,31,61,121,240,476,944,1872,3712,7362,14601,28958,
%T 57432,113904,225904,448034,888583,1762321,3495200,6932008,13748208,
%U 27266738,54077957,107252486,212713209,421872826,836697836,1659417786,3291113315,6527245245,12945446241,25674625681
%N Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.
%C 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
%C Row 4 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, and row 3 being A176503.
%H Alois P. Heinz, <a href="/A194628/b194628.txt">Table of n, a(n) for n = 1..1000</a>
%H 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.
%F a(n) = A294775(n-1,4). - _Alois P. Heinz_, Nov 08 2017
%e From _Joerg Arndt_, Dec 18 2012: (Start)
%e 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):
%e [ 1] [ 1 1 1 1 1 1 ]
%e [ 2] [ 1 1 1 1 2 ]
%e [ 3] [ 1 1 1 2 1 ]
%e [ 4] [ 1 1 1 3 ]
%e [ 5] [ 1 1 2 1 1 ]
%e [ 6] [ 1 1 2 2 ]
%e [ 7] [ 1 1 3 1 ]
%e [ 8] [ 1 1 4 ]
%e [ 9] [ 1 2 1 1 1 ]
%e [10] [ 1 2 1 2 ]
%e [11] [ 1 2 2 1 ]
%e [12] [ 1 2 3 ]
%e [13] [ 1 3 1 1 ]
%e [14] [ 1 3 2 ]
%e [15] [ 1 4 1 ]
%e [16] [ 1 5 ]
%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[4n - 3, 1, 5];
%t Array[a, 40] (* _Jean-François Alcover_, Jul 21 2018, after _Alois P. Heinz_ *)
%o (PARI) /* see A002572, set t=5 */
%Y Cf. A002572, A176485, A176503, A294775.
%K nonn
%O 1,4
%A _Jonathan Vos Post_, Aug 30 2011
%E Terms beyond a(20)=113904 added by _Joerg Arndt_, Dec 18 2012
%E Invalid empirical g.f. removed by _Alois P. Heinz_, Nov 08 2017
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