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

%I A194631 #24 Feb 15 2020 23:48:51
%S A194631 1,1,1,2,4,8,16,32,64,128,255,509,1017,2032,4060,8112,16208,32384,
%T A194631 64704,129280,258304,516098,1031177,2060318,4116568,8225008,16433776,
%U A194631 32835104,65605376,131081216,261903618,523290119,1045547025,2089029664,4173934632,8339628016
%N A194631 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.
%C A194631 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
%C A194631 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.
%H A194631 Alois P. Heinz, <a href="/A194631/b194631.txt">Table of n, a(n) for n = 1..1000</a>
%H A194631 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.
%F A194631 a(n) = A294775(n-1,7). - _Alois P. Heinz_, Nov 08 2017
%t A194631 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 A194631 a[n_] := b[7n-6, 1, 8];
%t A194631 Array[a, 40] (* _Jean-François Alcover_, Jul 21 2018, after _Alois P. Heinz_ *)
%o A194631 (PARI) /* see A002572, set t=8 */
%Y A194631 Cf. A002572, A176485, A176503, A194628, A194629, A194630, A294775.
%K A194631 nonn
%O A194631 1,4
%A A194631 _Jonathan Vos Post_, Aug 30 2011
%E A194631 Terms beyond a(20)=129280 added by _Joerg Arndt_, Dec 18 2012