cp's OEIS Frontend

A176503 Leading column of triangle in A176463.

%I A176503 #30 Mar 21 2019 17:18:13
%S A176503 1,1,1,2,4,8,15,29,57,112,220,432,848,1666,3273,6430,12632,24816,
%T A176503 48754,95783,188177,369696,726312,1426930,2803381,5507590,10820345,
%U A176503 21257915,41763825,82050242,161197933,316693445,622183778,1222357651,2401474098,4717995460
%N A176503 Leading column of triangle in A176463.
%C A176503 a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1), see example.  [_Joerg Arndt_, Dec 18 2012]
%H A176503 Alois P. Heinz, <a href="/A176503/b176503.txt">Table of n, a(n) for n = 1..1000</a>
%H A176503 Christian Elsholtz, Clemens Heuberger, Daniel Krenn, <a href="https://arxiv.org/abs/1901.11343">Algorithmic counting of nonequivalent compact Huffman codes</a>, arXiv:1901.11343 [math.CO], 2019.
%H A176503 Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="http://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
%F A176503 a(n) = A294775(n-1,3). - _Alois P. Heinz_, Nov 08 2017
%e A176503 From _Joerg Arndt_, Dec 18 2012: (Start)
%e A176503 There are a(6+1)=15 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1):
%e A176503 [ 1]  [ 1 1 1 1 1 1 ]
%e A176503 [ 2]  [ 1 1 1 1 2 ]
%e A176503 [ 3]  [ 1 1 1 2 1 ]
%e A176503 [ 4]  [ 1 1 1 3 ]
%e A176503 [ 5]  [ 1 1 2 1 1 ]
%e A176503 [ 6]  [ 1 1 2 2 ]
%e A176503 [ 7]  [ 1 1 3 1 ]
%e A176503 [ 8]  [ 1 1 4 ]
%e A176503 [ 9]  [ 1 2 1 1 1 ]
%e A176503 [10]  [ 1 2 1 2 ]
%e A176503 [11]  [ 1 2 2 1 ]
%e A176503 [12]  [ 1 2 3 ]
%e A176503 [13]  [ 1 3 1 1 ]
%e A176503 [14]  [ 1 3 2 ]
%e A176503 [15]  [ 1 4 1 ]
%e A176503 (End)
%t A176503 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 A176503 a[n_] := b[3n-2, 1, 4];
%t A176503 Array[a, 40] (* _Jean-François Alcover_, Jul 21 2018, after _Alois P. Heinz_ *)
%o A176503 (PARI)
%o A176503 /* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
%o A176503 N=66;  q='q+O('q^N);
%o A176503 t=4;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
%o A176503 L=2 + 2*ceil( log(N) / log(t) );
%o A176503 f(k) = (1-t^k)/(1-t);
%o A176503 la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
%o A176503 nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
%o A176503 dn=sum(j=0, L, (-1)^j * la(j) );
%o A176503 gf = nm / dn;
%o A176503 Vec( gf )
%o A176503 /* _Joerg Arndt_, Dec 27 2012 */
%Y A176503 Cf. A176463, A194628 - A194633, A294775.
%K A176503 nonn
%O A176503 1,4
%A A176503 _N. J. A. Sloane_, Dec 07 2010
%E A176503 Added terms beyond a(13)=848, _Joerg Arndt_, Dec 18 2012