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A176485
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First column of triangle in A176452.
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11
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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
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OFFSET
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1,4
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COMMENTS
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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]
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LINKS
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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.
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FORMULA
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EXAMPLE
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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)
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MATHEMATICA
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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];
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PROG
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(PARI)
/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66; q='q+O('q^N);
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 )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012.
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STATUS
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approved
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