A194631 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1017, 2032, 4060, 8112, 16208, 32384, 64704, 129280, 258304, 516098, 1031177, 2060318, 4116568, 8225008, 16433776, 32835104, 65605376, 131081216, 261903618, 523290119, 1045547025, 2089029664, 4173934632, 8339628016

Offset: 1

Author

Jonathan Vos Post, Aug 30 2011

Keywords

• nonn

Comments

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 8*p(k+1). - Joerg Arndt, Dec 18 2012

Row 7 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, and row 6 being A194630.

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:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Crossrefs

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

Programs

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[7n-6, 1, 8];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI

/* see A002572, set t=8 */

Formula

a(n) = A294775(n-1,7). - Alois P. Heinz, Nov 08 2017

Extensions

Terms beyond a(20)=129280 added by Joerg Arndt, Dec 18 2012