A247713 - OEIS

%I #12 Feb 06 2017 18:56:06

%S 1,1,5,52,4,451,48,2,3498,484,24,23502,4136,300,12,173611,37674,3262,

%T 142,1323447,335388,35938,1964,44,9920654,2892492,365458,25752,986,12,

%U 73573634,24266128,3544842,298200,15002,400,6,545170514,200531918,33123244,3236018,198380,7546,164,2

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

%C Sum_{k>0} k * T(n,k) = A247746(n).

%H Alois P. Heinz, <a href="/A247713/b247713.txt">Rows n = 0..170, flattened</a>

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

%e T(3,1) = 4:

%e ._____.        ._____.

%e |___. |        |   ._|

%e |_. | |        |___| |

%e | | |_|        | .___|

%e | |___|        |_|   |

%e |_____| (*2)   |_____| (*2)  .

%e Triangle T(n,k) begins:

%e 00 :        1;

%e 01 :        1;

%e 02 :        5;

%e 03 :       52,        4;

%e 04 :      451,       48,       2;

%e 05 :     3498,      484,      24;

%e 06 :    23502,     4136,     300,     12;

%e 07 :   173611,    37674,    3262,    142;

%e 08 :  1323447,   335388,   35938,   1964,    44;

%e 09 :  9920654,  2892492,  365458,  25752,   986,  12;

%e 10 : 73573634, 24266128, 3544842, 298200, 15002, 400, 6;

%Y Row sums give A174249 or A233427(n,5).

%Y Column k=0 gives A247777.

%Y Cf. A247746.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, Sep 23 2014