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A350637 Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.

%I A350637 #39 Mar 10 2022 04:56:22
%S A350637 1,4,1,8,4,1,15,8,4,1,21,15,8,4,1,33,21,15,8,4,1,41,33,21,15,8,4,1,56,
%T A350637 41,33,21,15,8,4,1,69,56,41,33,21,15,8,4,1,87,69,56,41,33,21,15,8,4,1,
%U A350637 99,87,69,56,41,33,21,15,8,4,1,127,99,87,69,56,41,33,21,15,8,4,1
%N A350637 Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.
%C A350637 T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the stepped pyramid with n levels described in A245092 (see example).
%H A350637 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>
%F A350637 T(n,k) = A024916(A004736(n,k)).
%F A350637 T(n,k) = T(n,k) = A024916(n-k+1).
%F A350637 T(n,k) = Sum_{j=1..n} A272172(j,k).
%e A350637 Triangle begins:
%e A350637     1;
%e A350637     4,  1;
%e A350637     8,  4,  1;
%e A350637    15,  8,  4,  1;
%e A350637    21, 15,  8,  4,  1;
%e A350637    33, 21, 15,  8,  4,  1;
%e A350637    41, 33, 21, 15,  8,  4,  1;
%e A350637    56, 41, 33, 21, 15,  8,  4,  1;
%e A350637    69, 56, 41, 33, 21, 15,  8,  4,  1;
%e A350637    87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
%e A350637    99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
%e A350637   127, 99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
%e A350637 ...
%e A350637 For n = 9 the lateral view and top view of the stepped pyramid described in A245092 look as shown below:
%e A350637                         _
%e A350637      9        1        |_|_
%e A350637      8        4        |_ _|_
%e A350637      7        8        |_ _|_|_
%e A350637      6       15        |_ _ _| |_
%e A350637      5       21        |_ _ _|_ _|_
%e A350637      4       33        |_ _ _ _| | |_
%e A350637      3       41        |_ _ _ _|_|_ _|_
%e A350637      2       56        |_ _ _ _ _|_|_  |_
%e A350637      1       69        |_ _ _ _ _|_ _|_ _|
%e A350637 .
%e A350637    Level   Row 9         Lateral view of
%e A350637      k     T(9,k)      the stepped pyramid
%e A350637 .
%e A350637                         _ _ _ _ _ _ _ _ _
%e A350637                        |_| | | | | | | | |
%e A350637                        |_ _|_| | | | | | |
%e A350637                        |_ _|  _|_| | | | |
%e A350637                        |_ _ _|    _|_| | |
%e A350637                        |_ _ _|  _|  _ _|_|
%e A350637                        |_ _ _ _|  _| |
%e A350637                        |_ _ _ _| |_ _|
%e A350637                        |_ _ _ _ _|
%e A350637                        |_ _ _ _ _|
%e A350637 .
%e A350637                            Top view of
%e A350637                        the stepped pyramid
%e A350637 .
%e A350637 For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the stepped pyramid, so T(9,1) = 69.
%e A350637 For n = 9 and k = 9 there is only one cubic cell in the level k = 9 (the top) of the stepped pyramid, so T(9,9) = 1.
%e A350637 The volume of the stepped pyramid (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000027 hence it's equal to A175254(9) = 248, equaling the sum of the 9th row of triangle.
%t A350637 Join@@Array[Reverse@Array[Sum[#-Mod[#,m],{m,#}]&,#]&,12] (* _Giorgos Kalogeropoulos_, Jan 12 2022 *)
%o A350637 (PARI) row(n) = Vecrev(vector(n, k, sum(i=1, k, k\i*i))); \\ _Michel Marcus_, Jan 22 2022
%Y A350637 Column k gives A024916 starting in row k.
%Y A350637 Row sums give A175254.
%Y A350637 Cf. A340423 (analog for the tower described in A221529).
%Y A350637 Cf. A000027, A000203, A004736, A196020, A235791, A236104, A237591, A237593, A245092, A262626, A272172.
%K A350637 nonn,tabl
%O A350637 1,2
%A A350637 _Omar E. Pol_, Jan 09 2022