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%I #13 Feb 06 2017 18:59:10
%S 1,1,5,56,461,32,8,3558,368,80,23966,3256,696,24,8,178127,29564,6558,
%T 360,80,1362597,266672,61858,4852,770,24,8,10194184,2361632,581452,
%U 58732,8890,384,80,75684682,20056764,5220634,632044,97174,5968,914,24,8
%N Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape W; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/2)*2) read by rows.
%C Sum_{k>0} k * T(n,k) = A247743(n).
%H Alois P. Heinz, <a href="/A247710/b247710.txt">Rows n = 0..145, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%e T(4,2) = 8:
%e ._______. ._______. ._______.
%e | ._____| |_. |_. | | ._____|
%e |_| ._| | | |_. | | |_| ._| |
%e | ._| ._| | | |_| | | ._| | |
%e |_|___| | | |_. |_| |_| ._| |
%e |_______| (*2) |___|___| (*2) |___|___| (*4)
%e Triangle T(n,k) begins:
%e 00 : 1;
%e 01 : 1;
%e 02 : 5;
%e 03 : 56;
%e 04 : 461, 32, 8;
%e 05 : 3558, 368, 80;
%e 06 : 23966, 3256, 696, 24, 8;
%e 07 : 178127, 29564, 6558, 360, 80;
%e 08 : 1362597, 266672, 61858, 4852, 770, 24, 8;
%e 09 : 10194184, 2361632, 581452, 58732, 8890, 384, 80;
%e 10 : 75684682, 20056764, 5220634, 632044, 97174, 5968, 914, 24, 8;
%Y Row sums give A174249 or A233427(n,5).
%Y Column k=0 gives A247774.
%Y Cf. A247743.
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Sep 23 2014