A194629 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

1, 1, 1, 2, 4, 8, 16, 32, 63, 125, 249, 496, 988, 1968, 3920, 7808, 15552, 30978, 61705, 122910, 244824, 487664, 971376, 1934880, 3854082, 7676935, 15291665, 30459424, 60672040, 120852464, 240725680, 479500802, 955116293, 1902493446, 3789571321, 7548436410

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) <= 6*p(k+1). - Joerg Arndt, Dec 18 2012

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

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-1,5). - 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[5n-4, 1, 6];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

Prog

(PARI) /* see A002572, set t=6 */

Crossrefs

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

Sequence in context: A239558 A239559 A001592 * A251710 A217832 A251740

Adjacent sequences: A194626 A194627 A194628 * A194630 A194631 A194632

Keyword

nonn

Author

Jonathan Vos Post, Aug 30 2011

Extensions

Terms beyond a(20)=122910 added by Joerg Arndt, Dec 18 2012

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

Status

approved