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A194632
Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.
3
1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2041, 4080, 8156, 16304, 32592, 65152, 130240, 260352, 520448, 1040384, 2079746, 4157449, 8310814, 16613464, 33210608, 66388592, 132711968, 265293568, 530326528, 1060132096, 2119222786, 4236363783, 8468566033
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) <= 9*p(k+1). - Joerg Arndt, Dec 18 2012
Row 8 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, row 5 being A194629, row 6 being A194630, and row 7 being A194631.
LINKS
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-1,8). - Alois P. Heinz, Nov 08 2017
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[8n-7, 1, 9];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)
PROG
(PARI) /* see A002572, set t=9 */
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Aug 30 2011
EXTENSIONS
Terms beyond a(20)=130240 added by Joerg Arndt, Dec 18 2012
STATUS
approved