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A338531 a(n) is the number of row-convex domino towers with n bricks (rows need not be offset).

%I A338531 #8 Dec 07 2020 02:47:00
%S A338531 1,4,16,61,225,815,2923,10428,37097,131776,467732,1659537,5886945,
%T A338531 20880912,74060619,262672473,931615218,3304121816,11718561425,
%U A338531 41561571533
%N A338531 a(n) is the number of row-convex domino towers with n bricks (rows need not be offset).
%C A338531 A domino tower is a stack of bricks, where (1) the bottom row is contiguous, and (2) each brick is supported from below by at least half of a brick. Note, that in this definition of domino towers, rows need not be offset by half a brick. The number of domino towers with n bricks is given by 4^(n-1).
%C A338531 In this sequence we want all rows to be convex, rather than just the bottom row.
%H A338531 T. M. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Brown/brown2.html">Convex domino towers</a>, J. of Integer Sequences, 20 (2017), p.17.
%F A338531 G.f.: G(x) := [ Sum_{l>0} z^l (z^3 T(3,l)+(2 z^2-1) T(2,l)+(2 z+1) T(1,l)) ] / (z^5 T(2,3)+(3 z-1) z^3 T(1,3)+(4 z^3-3 (z+1) z+1) T(1,2)) , where
%F A338531 T(i,j) := A(i)B(j)-A(j)B(i),
%F A338531 A(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)_n)/((z;z)_n)^2,
%F A338531 B(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)_n)/((z;z)_n)^2 * (l+n+Sum_{m=1,...,n} (3 z^m+1)/(1-z^(2 m))), and
%F A338531 (a;q)_n is the q-Pochhammer symbol
%e A338531 For n=2, the a(2) = 4 domino towers are:
%e A338531 +-------+-------+
%e A338531 |       |       |
%e A338531 +-------+-------+
%e A338531 +-------+
%e A338531 |       |
%e A338531 +---+---+---+
%e A338531     |       |
%e A338531     +-------+
%e A338531 +-------+
%e A338531 |       |
%e A338531 +-------+
%e A338531 |       |
%e A338531 +-------+
%e A338531     +-------+
%e A338531     |       |
%e A338531 +---+---+---+
%e A338531 |       |
%e A338531 +-------+
%e A338531 For n=4, the 4^(n-1)-a(n)=64-61=3 domino towers, which have non-convex rows are:
%e A338531 +-------+   +-------+
%e A338531 |       |   |       |
%e A338531 +-------+---+---+---+
%e A338531 |       |       |
%e A338531 +-------+-------+
%e A338531 +-------+   +-------+
%e A338531 |       |   |       |
%e A338531 +---+---+---+-------+
%e A338531     |       |       |
%e A338531     +-------+-------+
%e A338531 +-------+       +-------+
%e A338531 |       |       |       |
%e A338531 +---+---+---+---+---+---+
%e A338531     |       |       |
%e A338531     +-------+-------+
%t A338531 f[n_, l_] := (f[n, l] =
%t A338531 Sum[(3 - 2 i + 2 l) f[n - i, i], {i, 1, Min[n, l + 1]}]);
%t A338531 f[0, l_] := 1;
%t A338531 Table[Sum[f[n - l, l], {l, 1, n}], {n, 1, 20}]
%Y A338531 Cf. A275204.
%K A338531 nonn,more
%O A338531 1,2
%A A338531 _Alexander M. Haupt_, Nov 01 2020