A194630 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

1, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 505, 1008, 2012, 4016, 8016, 16000, 31936, 63744, 127234, 253961, 506910, 1011800, 2019568, 4031088, 8046112, 16060160, 32056322, 63984903, 127714833, 254920736, 508825640, 1015623664, 2027200176, 4046322176, 8076520194

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) <= 7*p(k+1). - Joerg Arndt, Dec 18 2012

Row 6 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, and row 5 being A194629.

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.

Formula

a(n) = A294775(n-1,6). - 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[6n-5, 1, 7];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

Prog

(PARI) /* see A002572, set t=7 */

Crossrefs

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

Sequence in context: A239560 A066178 A122189 * A251672 A251747 A251761

Adjacent sequences: A194627 A194628 A194629 * A194631 A194632 A194633

Keyword

nonn

Author

Jonathan Vos Post, Aug 30 2011

Extensions

Terms beyond a(20)=127234 added by Joerg Arndt, Dec 18 2012

Status

approved