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A379624 Triangle read by rows: T(n,k) is the number of free polyominoes with n cells and length k, n >= 1, k = 1..n.

%I A379624 #53 Feb 17 2025 22:37:18
%S A379624 1,0,1,0,1,1,0,1,3,1,0,0,8,3,1,0,0,8,21,5,1,0,0,7,59,36,5,1,0,0,3,137,
%T A379624 167,54,7,1,0,0,1,223,669,307,77,7,1,0,0,0,287,2089,1627,539,103,9,1,
%U A379624 0,0,0,255,5472,7126,3237,839,134,9,1,0,0,0,169,11919,27504,16706,5851,1271,168,11,1
%N A379624 Triangle read by rows: T(n,k) is the number of free polyominoes with n cells and length k, n >= 1, k = 1..n.
%C A379624 The length here is the longer of the two dimensions.
%H A379624 John Mason, <a href="/A379624/b379624.txt">Table of n, a(n) for n = 1..171</a> (first 18 rows)
%H A379624 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%e A379624 Triangle begins:
%e A379624   1;
%e A379624   0,  1;
%e A379624   0,  1,  1;
%e A379624   0,  1,  3,    1;
%e A379624   0,  0,  8,    3,     1;
%e A379624   0,  0,  8,   21,     5,     1;
%e A379624   0,  0,  7,   59,    36,     5,     1;
%e A379624   0,  0,  3,  137,   167,    54,     7,    1;
%e A379624   0,  0,  1,  223,   669,   307,    77,    7,    1;
%e A379624   0,  0,  0,  287,  2089,  1627,   539,  103,    9,   1;
%e A379624   0,  0,  0,  255,  5472,  7126,  3237,  839,  134,   9,   1;
%e A379624   0,  0,  0,  169, 11919, 27504, 16706, 5851, 1271, 168,  11,  1;
%e A379624   ...
%e A379624 Illustration for n = 5:
%e A379624 The free polyominoes with five cells are also called free pentominoes.
%e A379624 For k = 1 there are no free pentominoes of length 1, so T(5,1) = 0.
%e A379624 For k = 2 there are no free pentominoes of length 2, so T(5,2) = 0.
%e A379624 For k = 3 there are eight free pentominoes of length 3 as shown below, so T(5,3) = 8.
%e A379624    _ _     _ _       _ _     _ _ _     _         _           _       _ _
%e A379624   |_|_|   |_|_|    _|_|_|   |_|_|_|   |_|       |_|_       _|_|_    |_|_|
%e A379624   |_|_|   |_|_    |_|_|       |_|     |_|_ _    |_|_|_    |_|_|_|     |_|_
%e A379624   |_|     |_|_|     |_|       |_|     |_|_|_|     |_|_|     |_|       |_|_|
%e A379624 .
%e A379624 For k = 4 there are three free pentominoes of length 4 as shown below, so T(5,4) = 3.
%e A379624    _         _       _
%e A379624   |_|      _|_|    _|_|
%e A379624   |_|     |_|_|   |_|_|
%e A379624   |_|_    |_|       |_|
%e A379624   |_|_|   |_|       |_|
%e A379624 .
%e A379624 For k = 5 there is only one free pentomino of length 5 as shown below, so T(5,5) = 1.
%e A379624    _
%e A379624   |_|
%e A379624   |_|
%e A379624   |_|
%e A379624   |_|
%e A379624   |_|
%e A379624 .
%e A379624 Therefore the 5th row of the triangle is [0, 0, 8, 3, 1] and the row sum is A000105(5) = 12.
%e A379624 .
%Y A379624 Row sums give A000105(n).
%Y A379624 Column 1 gives A000007.
%Y A379624 Leading diagonal gives A000012.
%Y A379624 For free polyominoes of width k see A379623.
%Y A379624 Cf. A109613, A379627, A379628.
%K A379624 nonn,tabl
%O A379624 1,9
%A A379624 _Omar E. Pol_, Jan 07 2025
%E A379624 Terms a(37) and beyond from _Jinyuan Wang_, Jan 08 2025