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A247703 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape I; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A247703 #16 Feb 06 2017 18:42:05
%S A247703 1,0,1,4,0,1,47,8,0,1,394,94,12,0,1,2082,1608,282,32,0,2,15113,8812,
%T A247703 3452,512,58,0,3,111664,73863,22310,5962,790,96,0,4,789930,631700,
%U A247703 218608,45762,9374,1260,142,0,5,5388729,5157928,2067811,491868,81720,15272,1824,196,0,6
%N A247703 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape I; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A247703 Sum_{k>0} k * T(n,k) = A247736(n).
%H A247703 Alois P. Heinz, <a href="/A247703/b247703.txt">Rows n = 0..140, flattened</a>
%H A247703 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%e A247703 T(5,5) = 2:
%e A247703 ._._._._._.   ._________.
%e A247703 | | | | | |   |_________|
%e A247703 | | | | | |   |_________|
%e A247703 | | | | | |   |_________|
%e A247703 | | | | | |   |_________|
%e A247703 |_|_|_|_|_|   |_________| .
%e A247703 Triangle T(n,k) begins:
%e A247703 00 :      1;
%e A247703 01 :      0,      1;
%e A247703 02 :      4,      0,      1;
%e A247703 03 :     47,      8,      0,     1;
%e A247703 04 :    394,     94,     12,     0,    1;
%e A247703 05 :   2082,   1608,    282,    32,    0,    2;
%e A247703 06 :  15113,   8812,   3452,   512,   58,    0,   3;
%e A247703 07 : 111664,  73863,  22310,  5962,  790,   96,   0,  4;
%e A247703 08 : 789930, 631700, 218608, 45762, 9374, 1260, 142,  0,  5;
%Y A247703 Row sums give A174249 or A233427(n,5).
%Y A247703 Column k=0 gives A247767.
%Y A247703 Main diagonal gives A003520.
%Y A247703 Cf. A247736.
%K A247703 nonn,tabl
%O A247703 0,4
%A A247703 _Alois P. Heinz_, Sep 22 2014