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A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.

%I A091980 #43 Feb 18 2025 20:36:07
%S A091980 1,2,3,5,7,11,16,26,36,56,81,131,183,287,417,677,937,1457,2107,3407,
%T A091980 4759,7463,10843,17603,24373,37913,54838,88688,123892,194300,282310,
%U A091980 458330,634350,986390,1426440,2306540,3221844,5052452,7340712,11917232,16500522
%N A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.
%C A091980 The maximum is always obtained by taking i as the power of 2 nearest to n/2. - _Anna de Mier_, Mar 12 2012
%C A091980 a(n) is the number of (binary) max-heaps on n-1 elements from the set {0,1}. a(7) = 16: 000000, 100000, 101000, 101001, 110000, 110010, 110100, 110110, 111000, 111001, 111010, 111011, 111100, 111101, 111110, 111111. - _Alois P. Heinz_, Jul 09 2019
%D A091980 A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Graphs Combin., 28 (2012), 265-275.
%H A091980 Alois P. Heinz, <a href="/A091980/b091980.txt">Table of n, a(n) for n = 1..5652</a>
%H A091980 F. Disanto and N. A. Rosenberg, <a href="https://doi.org/10.1089/cmb.2016.0159">Enumeration of ancestral configurations for matching gene trees and species trees</a>, J. Comput. Biol. 24 (2017), 831-850. See Section 4.2.
%H A091980 A. de Mier and M. Noy, <a href="https://doi.org/10.1016/j.endm.2009.07.081">On the maximum number of cycles in outerplanar and series-parallel graphs</a>, Elect. Notes Discr. Math 34 (2009) 489-493
%H A091980 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Heap.html">Heap</a>
%H A091980 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_heap">Binary heap</a>
%F A091980 a(n) = 1 + max_{i=1..n-1} a(i)*a(n-i) for n > 1, a(1) = 1.
%F A091980 From _Alois P. Heinz_, Jul 09 2019: (Start)
%F A091980 a(n) = Sum_{k=0..n-1} A309049(n-1,k).
%F A091980 a(2^(n-1)) = A003095(n). (End)
%p A091980 b:= proc(n) option remember; `if`(n=0, 1, (g-> (f->
%p A091980       1+b(f)*b(n-1-f))(min(g-1, n-g/2)))(2^ilog2(n)))
%p A091980     end:
%p A091980 a:= n-> b(n-1):
%p A091980 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2019
%t A091980 a[n_] := a[n] = 1 + Max[Table[a[i] a[n-i], {i, n-1}]]; a[1] = 1;
%t A091980 Array[a, 50] (* _Jean-François Alcover_, Apr 30 2020 *)
%Y A091980 Cf. A003095, A056971, A309049.
%Y A091980 Partial differences give A168542.
%Y A091980 a(n) = A355108(n)+1.
%Y A091980 Column k=0 of A370484 and of A372640.
%K A091980 easy,nonn
%O A091980 1,2
%A A091980 _Franklin T. Adams-Watters_, Mar 15 2004