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%I #16 Oct 19 2022 20:10:28
%S 1,1,3,15,495,467775,448046589375,396822986774382287109375,
%T 316789536051348354157896789538095888519287109375
%N Number of matchings in the complete binary tree of n rows.
%C A recurrence is formed by considering the root vertex matched or unmatched and a(n-1) or a(n-2) matchings in the subtrees below.
%C unmatched matched matched
%C / \ / \\ // \
%C any any any matched matched any
%C / \ / \
%C any any any any
%C so:
%C a(n-1)^2 + 2 * a(n-1)*a(n-2)^2 = a(n)
%C The Jacobsthal product formula (below) follows from this recurrence by induction by substituting the products for a(n-1) and a(n-2) and using J(n+1) = J(n) + 2*J(n-1) (its recurrence in A001045).
%C The Jacobsthal product terms, with multiplicity, in a(n) are a subset of the terms in any bigger a(m), so a(n) divides any bigger a(m) and so in particular this is a divisibility sequence.
%C Asymptotically, a(n) ~ (1/2)*C^(2^n) where C = 1.537176.. = A338294. For growth power C, let c(n) = (2*a(n))^(1/2^n) so that C = lim_{n->oo} c(n). The Jacobsthal products formula gives log(c(n)) = log(2)/2^n + log(J(n+1))/2^n + Sum_{k=1..n} log(J(k))/2^k. Then discarding log(J(1)) = log(J(2)) = 0, and log(2)/2^n -> 0, and log(J(n+1))/2^n -> 0, leaves the terms of A242049 so that log(C) = A242049.
%C The asymptotic factor F = 1/2 is found by letting f(n) = a(n)/a(n-1)^2, so f(n) = J(n+1) / J(n) by the products formula, and f(n) = 2 + (-1)^n/J(n) -> 2 = 1/F. This factor makes no difference to the growth power C, since any F^(1/2^n) -> 1, but it brings the approximation closer to a(n) sooner.
%H Kevin Ryde, <a href="/A338293/b338293.txt">Table of n, a(n) for n = 0..12</a>
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/pari-vpar/index.html">vpar</a> examples/complete-binary-matchings.gp calculations and code in PARI/GP.
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = a(n-1)^2 + 2*a(n-1)*a(n-2)^2 starting a(0)=1 and a(1)=1.
%F a(n) = J(n+1) * J(n) * J(n-1)^2 * J(n-2)^4 * ... * J(1)^(2^(n-1)) where J(n) = (2^n - (-1)^n)/3 = A001045(n) is the Jacobsthal numbers.
%F A228607(n) = a(n+1)^2*(a(n+1)+3*a(n)^2). - _R. J. Mathar_, Jul 22 2022
%e n=0 rows is the empty tree and n=1 row is a single vertex. Both have only the empty matching so a(0) = a(1) = 1.
%e n=2 rows is a path-3 and has 3 matchings: first two vertices, last two, or the empty matching, so a(2) = 3.
%e Jacobsthal products formula:
%e a(4) = J(5) * J(4) * J(3)^2 * J(2)^4 * J(1)^8
%e = 11 * 5 * 3^2 * 1^4 * 1^8 = 495.
%o (PARI) a(n) = my(x=1,y=1); for(i=2,n, [x,y] = [x^2 + 2*x*y^2, x]); x;
%Y Cf. A001045 (Jacobsthal numbers), A338294 (growth power), A242049 (log of growth power).
%Y Cf. A076725 (independent sets), A158681 (Wiener index), A000975 (independence number and matching number).
%K nonn
%O 0,3
%A _Kevin Ryde_, Oct 21 2020
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