A360665 - OEIS

%I #38 Apr 20 2023 06:41:51

%S 0,0,0,0,1,-1,0,2,3,-3,0,3,7,6,-6,0,4,11,15,10,-10,0,5,15,24,26,15,

%T -15,0,6,19,33,42,40,21,-21,0,7,23,42,58,65,57,28,-28,0,8,27,51,74,90,

%U 93,77,36,-36,0,9,31,60,90,115,129,126,100,45,-45

%N Square array T(n, k) = k*((2*n-1)*k+1)/2 read by rising antidiagonals.

%F T(n,k) = T(n,k-1)+k^2.

%F T(n,n) = A081436(n-1).

%F T(n,n+1) = A059270(n).

%F T(n,n+4) = -3*A179297(n+4).

%F T(n+3,n) = A162254(n).

%F T(n+5,n) = 3*A101986(n).

%F From _Stefano Spezia_, Mar 31 2023: (Start)

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

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

%e By rows:

%e    0,   0,  -1,  -3,  -6,  -10,  -15,  -21,  -28, ...   = -A161680

%e    0,   1,   3,   6,  10,   15,   21,   28,   36, ...   =  A000217

%e    0,   2,   7,  15,  26,   40,   57,   77,  100, ...   =  A005449

%e    0,   3,  11,  24,  42,   65,   93,  126,  164, ...   =  A005475

%e    0,   4,  15,  33,  58,   90,  129,  175,  228, ...   =  A022265

%e    0,   5,  19,  42,  74,  115,  165,  224,  292, ...   =  A022267

%e    0,   6,  23,  51,  90,  140,  201,  273,  356, ...   =  A022269

%e    ... .

%t T[n_, k_] := ((2*n - 1)*k^2 + k)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Mar 31 2023 *)

%o (PARI) T(n, k) = ((2*n-1)*k^2+k)/2 \\ _Thomas Scheuerle_, Mar 17 2023

%Y Cf. Antidiagonal sums: A034827(n+1).

%Y Cf. A000217, A000290, A005449, A005475, A022265.

%Y Cf. A022267, A022269, A059270, A081436, A101986.

%Y Cf. A161680, A162254, A179297, A360962, A361226.

%K sign,tabl,easy

%O 0,8

%A _Paul Curtz_, Mar 17 2023