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A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

%I A121682 #21 Nov 17 2022 06:25:57
%S A121682 1,6,4,27,21,9,124,100,52,16,645,525,285,105,25,3906,3186,1746,666,
%T A121682 186,36,27391,22351,12271,4711,1351,301,49,219192,178872,98232,37752,
%U A121682 10872,2472,456,64,1972809,1609929,884169,339849,97929,22329,4185,657,81,19728190,16099390,8841790,3398590,979390,223390,41950,6670,910,100
%N A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.
%C A121682 The first column is A030297 = a(n) = n*(n+a(n-1)). The main diagonal are the squares A000290 = n^2. The first lower diagonal (6,21,52,...) is A069778 = q-factorial numbers 3!_q. See also A121662.
%D A121682 T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%e A121682 Triangle begins:
%e A121682       1
%e A121682       6     4
%e A121682      27    21     9
%e A121682     124   100    52   16
%e A121682     645   525   285  105  25
%e A121682    3906  3186  1746  666  186  36
%e A121682   27391 22351 12271 4711 1351 301 49
%e A121682   ...
%p A121682 T:= proc(i, j) option remember;
%p A121682       `if`(j<1 or j>i, 0, (T(i-1, j)+i)*i)
%p A121682     end:
%p A121682 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Jun 22 2022
%t A121682 T[n_, k_] /; 1 <= k <= n := T[n, k] = (T[n-1, k]+n)*n;
%t A121682 T[_, _] = 0;
%t A121682 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 17 2022 *)
%o A121682 (Python)
%o A121682 def T(i, j): return (T(i-1, j)+i)*i if 1 <= j <= i else 0
%o A121682 print([T(r, c) for r in range(1, 11) for c in range(1, r+1)]) # _Michael S. Branicky_, Jun 22 2022
%Y A121682 Cf. A030297, A000290, A069778, A121662.
%Y A121682 Row sums give A337001.
%K A121682 nonn,tabl
%O A121682 1,2
%A A121682 _Thomas Wieder_, Aug 15 2006
%E A121682 Edited by _N. J. A. Sloane_, Sep 15 2006
%E A121682 Formula in name corrected by _Alois P. Heinz_, Jun 22 2022