cp's OEIS Frontend

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

%I A194633 #31 May 31 2021 17:07:27
%S A194633 1,1,1,2,4,8,16,32,64,128,256,512,1023,2045,4089,8176,16348,32688,
%T A194633 65360,130688,261312,522496,1044736,2088960,4176896,8351746,16699401,
%U A194633 33390622,66764888,133497072,266928752,533726752,1067192064,2133861376,4266677504,8531265024
%N A194633 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.
%C A194633 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]
%C A194633 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.
%H A194633 Alois P. Heinz, <a href="/A194633/b194633.txt">Table of n, a(n) for n = 1..1000</a>
%H A194633 Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="https://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
%F A194633 a(n) = A294775(n-1,9).
%t A194633 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]}]]];
%t A194633 a[n_] := b[9n-8, 1, 10];
%t A194633 Array[a, 40] (* _Jean-François Alcover_, Jul 21 2018, after _Alois P. Heinz_ *)
%o A194633 (PARI) /* see A002572, set t=10 */
%Y A194633 Cf. A002572, A176485, A176503, A194628 - A194632, A294775.
%K A194633 nonn
%O A194633 1,4
%A A194633 _Jonathan Vos Post_, Aug 30 2011
%E A194633 Added terms beyond a(20)=130688, _Joerg Arndt_, Dec 18 2012
%E A194633 Invalid empirical g.f. removed by _Alois P. Heinz_, Nov 08 2017