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A341148 Triangle read by rows: T(n,k) is number of cubes in the k-th vertical slice of the polycube called "tower" described in A221529 where n is the longest side of its base, 1 <= k <= n.

%I A341148 #48 Jun 09 2022 02:30:20
%S A341148 1,2,2,4,3,2,7,6,4,3,12,10,7,3,3,19,17,12,9,5,4,30,26,20,13,8,4,4,45,
%T A341148 41,31,23,16,10,5,5,67,60,48,34,25,15,11,5,5,97,89,71,55,39,28,17,12,
%U A341148 6,6,139,127,104,78,60,40,28,17,11,6,6,195,181,149,118,89,65,45,32,21,15,7,7
%N A341148 Triangle read by rows: T(n,k) is number of cubes in the k-th vertical slice of the polycube called "tower" described in A221529 where n is the longest side of its base, 1 <= k <= n.
%C A341148 The row sums of triangle give A066186 because the correspondence divisor/part. For more information see A338156.
%C A341148 For further information about the tower see A221529.
%e A341148 Triangle begins:
%e A341148     1;
%e A341148     2,   2;
%e A341148     4,   3,   2;
%e A341148     7,   6,   4,   3;
%e A341148    12,  10,   7,   3,  3;
%e A341148    19,  17,  12,   9,  5,  4;
%e A341148    30,  26,  20,  13,  8,  4,  4;
%e A341148    45,  41,  31,  23, 16, 10,  5,  5;
%e A341148    67,  60,  48,  34, 25, 15, 11,  5,  5;
%e A341148    97,  89,  71,  55, 39, 28, 17, 12,  6,  6;
%e A341148   139, 127, 104,  78, 60, 40, 28, 17, 11,  6,  6;
%e A341148   195, 181, 149, 118, 89, 65, 45, 32, 21, 15,  7,  7;
%e A341148 ...
%e A341148 Illustration of initial terms:
%e A341148               Top view
%e A341148   n   k       of the tower       Heights        T(n,k)
%e A341148                _
%e A341148   1   1       |_|                1                 1
%e A341148 .              _ _
%e A341148   2   1       |   |              1 1               2
%e A341148   2   2       |_ _|              1 1               2
%e A341148 .              _ _ _
%e A341148   3   1       |_|   |            2 1 1             4
%e A341148   3   2       |    _|            1 1 1             3
%e A341148   3   3       |_ _|              1 1               2
%e A341148 .              _ _ _ _
%e A341148   4   1       |_| |   |          3 2 1 1           7
%e A341148   4   2       |_ _|   |          2 2 1 1           6
%e A341148   4   3       |      _|          1 1 1 1           4
%e A341148   4   4       |_ _ _|            1 1 1             3
%e A341148 .              _ _ _ _ _
%e A341148   5   1       |_| | |   |        5 3 2 1 1        12
%e A341148   5   2       |_ _|_|   |        3 3 2 1 1        10
%e A341148   5   3       |_ _|  _ _|        2 2 1 1 1         7
%e A341148   5   4       |     |            1 1 1             3
%e A341148   5   5       |_ _ _|            1 1 1             3
%e A341148 .              _ _ _ _ _ _
%e A341148   6   1       |_| | | |   |      7 5 3 2 1 1      19
%e A341148   6   2       |_ _|_| |   |      5 5 3 2 1 1      17
%e A341148   6   3       |_ _|  _|   |      3 3 2 2 1 1      12
%e A341148   6   4       |_ _ _|    _|      2 2 2 1 1 1       9
%e A341148   6   5       |        _|        1 1 1 1 1         5
%e A341148   6   6       |_ _ _ _|          1 1 1 1           4
%e A341148 .
%e A341148 The levels of the terraces of the tower are the partition numbers A000041 starting from the base.
%e A341148 Note that the top view of the tower is essentially the same as the top view of the stepped pyramid described in A245092 except that in the tower both the symmetric representation of sigma(n) and the symmetric representation of sigma(n-1) are unified in the level 1 of the structure because the first two partitions numbers A000041 are [1, 1].
%Y A341148 Column 1 gives A000070.
%Y A341148 Leading diagonal gives A080513.
%Y A341148 Row sums give A066186.
%Y A341148 Cf. A000041, A024916, A236104, A237270, A237271, A237593, A221529, A245092, A262626, A336811, A336812, A338156, A340035.
%K A341148 nonn,tabl
%O A341148 1,2
%A A341148 _Omar E. Pol_, Feb 06 2021