A304677 - OEIS

%I #8 May 19 2018 15:36:11

%S 1,1,4,9,27,60,170,377,996,2288,5715,13002,32321,72864,175137,400039,

%T 943454,2133159,4993737,11236889,25995341,58480330,133650880,

%U 299347432,681346296,1519116099,3427954877,7631479391,17122129103,37958987956,84819325972,187405201004

%N Total number of tilings of Ferrers-Young diagrams using dominoes and monominoes summed over all partitions of n.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>

%p h:= proc(l, f) option remember; local k; if min(l[])>0 then

%p      `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))

%p     else for k from nops(l) while l[k]>0 by -1 do od; h(subsop(k=1, l), f)+

%p         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+

%p         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)

%p       fi

%p     end:

%p g:= l-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))):

%p b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)

%p                   +b(n-i, min(n-i, i), [l[], i])):

%p a:= n-> b(n$2, []):

%p seq(a(n), n=0..23);

%Y Cf. A304662, A304680.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 16 2018