A194633 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4089, 8176, 16348, 32688, 65360, 130688, 261312, 522496, 1044736, 2088960, 4176896, 8351746, 16699401, 33390622, 66764888, 133497072, 266928752, 533726752, 1067192064, 2133861376, 4266677504, 8531265024

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) <= 10*p(k+1). [Joerg Arndt, Dec 18 2012]

Row 9 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, row 7 being A194631, and row 8 being A194632.

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-1,9).

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[9n-8, 1, 10];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

Prog

(PARI) /* see A002572, set t=10 */

Crossrefs

Cf. A002572, A176485, A176503, A194628 - A194632, A294775.

Sequence in context: A234591 A122265 A339073 * A243088 A113699 A113010

Adjacent sequences: A194630 A194631 A194632 * A194634 A194635 A194636

Keyword

nonn

Author

Jonathan Vos Post, Aug 30 2011

Extensions

Added terms beyond a(20)=130688, Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017

Status

approved