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A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

%I A360962 #44 Feb 22 2024 02:15:45
%S A360962 0,0,1,0,4,5,0,7,17,12,0,10,29,39,22,0,13,41,66,70,35,0,16,53,93,118,
%T A360962 110,51,0,19,65,120,166,185,159,70,0,22,77,147,214,260,267,217,92,0,
%U A360962 25,89,174,262,335,375,364,284,117,0,28,101,201,310,410,483,511,476,360,145
%N A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.
%C A360962 The main diagonal is A024394.
%C A360962 The antidiagonals sums are A000537.
%F A360962 Take successively sequences n*(3*n-1)/2, n*(9*n-1)/2, n*(15*n-1)/2, n*(21*n-1)/2, ... listed in the EXAMPLE section.
%F A360962 From _Stefano Spezia_, Feb 21 2024: (Start)
%F A360962 G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).
%F A360962 E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)
%e A360962 The rows are:
%e A360962   0  1  5  12  22  35  51  70 ... = A000326
%e A360962   0  4 17  39  70 110 159 217 ... = A022266
%e A360962   0  7 29  66 118 185 267 364 ... = A022272
%e A360962   0 10 41  93 166 260 375 511 ... = A022278
%e A360962   0 13 53 120 214 335 483 658 ... = A022284
%e A360962   ... .
%e A360962 Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).
%e A360962 Difference between two consecutive rows are: A033428.
%e A360962 This square array read by antidiagonals leads to the triangle
%e A360962   0
%e A360962   0  1
%e A360962   0  4  5
%e A360962   0  7 17 12
%e A360962   0 10 29 39  22
%e A360962   0 13 41 66  70  35
%e A360962   0 16 53 93 118 110 51
%e A360962   ... .
%p A360962 T:= (n,k)-> k*(k*(3+6*n)-1)/2:
%p A360962 seq(seq(T(d-k,k), k=0..d), d=0..10);  # _Alois P. Heinz_, Feb 28 2023
%t A360962 T[n_, k_] := ((6*n + 3)*k - 1)*k/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 27 2023 *)
%o A360962 (PARI) T(n,k) = k*((3+6*n)*k-1)/2; \\ _Michel Marcus_, Feb 27 2023
%Y A360962 Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.
%Y A360962 Cf. A033428.
%Y A360962 Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.
%K A360962 nonn,tabl
%O A360962 0,5
%A A360962 _Paul Curtz_, Feb 27 2023