A360962 - OEIS

%I #44 Feb 22 2024 02:15:45

%S 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 110,51,0,19,65,120,166,185,159,70,0,22,77,147,214,260,267,217,92,0,

%U 25,89,174,262,335,375,364,284,117,0,28,101,201,310,410,483,511,476,360,145

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

%C The main diagonal is A024394.

%C The antidiagonals sums are A000537.

%F 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 From _Stefano Spezia_, Feb 21 2024: (Start)

%F G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).

%F E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)

%e The rows are:

%e   0  1  5  12  22  35  51  70 ... = A000326

%e   0  4 17  39  70 110 159 217 ... = A022266

%e   0  7 29  66 118 185 267 364 ... = A022272

%e   0 10 41  93 166 260 375 511 ... = A022278

%e   0 13 53 120 214 335 483 658 ... = A022284

%e   ... .

%e Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).

%e Difference between two consecutive rows are: A033428.

%e This square array read by antidiagonals leads to the triangle

%e   0

%e   0  1

%e   0  4  5

%e   0  7 17 12

%e   0 10 29 39  22

%e   0 13 41 66  70  35

%e   0 16 53 93 118 110 51

%e   ... .

%p T:= (n,k)-> k*(k*(3+6*n)-1)/2:

%p seq(seq(T(d-k,k), k=0..d), d=0..10);  # _Alois P. Heinz_, Feb 28 2023

%t 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 (PARI) T(n,k) = k*((3+6*n)*k-1)/2; \\ _Michel Marcus_, Feb 27 2023

%Y Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.

%Y Cf. A033428.

%Y Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.

%K nonn,tabl

%O 0,5

%A _Paul Curtz_, Feb 27 2023