A194632 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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 */

Crossrefs

Cf. A002572, A176485, A176503, A194628, A194629, A194630, A194631, A294775.

Sequence in context: A234590 A104144 A258800 * A282584 A251759 A243087

Adjacent sequences: A194629 A194630 A194631 * A194633 A194634 A194635

Keyword

nonn

Author

Jonathan Vos Post, Aug 30 2011

Extensions

Terms beyond a(20)=130240 added by Joerg Arndt, Dec 18 2012

Status

approved