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A247712 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Y; triangle T(n,k), n>=0, read by rows.

%I A247712 #19 Feb 06 2017 18:57:35
%S A247712 1,1,5,44,12,321,136,44,2404,1160,404,24,14,14692,9380,3388,392,90,8,
%T A247712 98831,78492,30834,5724,748,60,684729,631020,292250,74016,13280,1428,
%U A247712 58,4642752,4944856,2628566,788284,171368,25648,3648,228,4
%N A247712 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Y; triangle T(n,k), n>=0, read by rows.
%C A247712 Sum_{k>0} k * T(n,k) = A247745(n).
%C A247712 T(10*n,10*n) = 10^n = A011557(n).
%H A247712 Alois P. Heinz, <a href="/A247712/b247712.txt">Rows n = 0..145, flattened</a>
%H A247712 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%e A247712 T(3,1) = 12:
%e A247712 ._____.        ._____.        ._____.
%e A247712 | |_. |        |_.   |        | |_. |
%e A247712 | ._| |        | |___|        | ._| |
%e A247712 | | | |        | ._| |        | |___|
%e A247712 |_| |_|        | |   |        |_|   |
%e A247712 |_____| (*4)   |_|___| (*4)   |_____| (*4)  .
%e A247712 T(10,10) = 10:
%e A247712 .___________________.
%e A247712 |_. .___| |___. ._| |
%e A247712 | |_| |_______|_|_. |
%e A247712 | |_______|___. ._| |
%e A247712 | ._| |___. ._|_| |_|
%e A247712 |_|_______|_|_______| ... .
%e A247712 Triangle T(n,k) begins:
%e A247712 00 :      1;
%e A247712 01 :      1;
%e A247712 02 :      5;
%e A247712 03 :     44,     12;
%e A247712 04 :    321,    136,     44;
%e A247712 05 :   2404,   1160,    404,    24,    14;
%e A247712 06 :  14692,   9380,   3388,   392,    90,    8;
%e A247712 07 :  98831,  78492,  30834,  5724,   748,   60;
%e A247712 08 : 684729, 631020, 292250, 74016, 13280, 1428, 58;
%Y A247712 Row sums give A174249 or A233427(n,5).
%Y A247712 Column k=0 gives A247776.
%Y A247712 Cf. A011557, A247745.
%K A247712 nonn,tabf
%O A247712 0,3
%A A247712 _Alois P. Heinz_, Sep 23 2014