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A361226 Square array T(n,k) = k*((1+2*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

%I A361226 #54 Feb 22 2024 02:03:10
%S A361226 0,0,0,0,1,1,0,2,5,3,0,3,9,12,6,0,4,13,21,22,10,0,5,17,30,38,35,15,0,
%T A361226 6,21,39,54,60,51,21,0,7,25,48,70,85,87,70,28,0,8,29,57,86,110,123,
%U A361226 119,92,36,0,9,33,66,102,135,159,168,156,117,45
%N A361226 Square array T(n,k) = k*((1+2*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.
%C A361226 The main diagonal is A002414.
%C A361226 The first upper diagonal is A160378(n+1).
%C A361226 The antidiagonals sums are A034827(n+2).
%C A361226 b(n) = (A034827(n+3) = 0, 2, 10, 30, 70, ...) - (A002414(n) = 0, 1, 9, 30, 70, ...) = 0, 1, 1, 0, 0, 5, 21, 56, ... .
%C A361226 b(n+2) = A299120(n). b(n+4) = A033275(n). b(n+4) - b(n) = A002492(n).
%F A361226 Take successively sequences n*(n-1)/2, n*(3*n-1)/2, n*(5*n-1)/2, ... listed in the EXAMPLE section.
%F A361226 G.f.: y*(x + y)/((1 - y)^3*(1 - x)^2). - _Stefano Spezia_, Mar 06 2023
%F A361226 E.g.f.: exp(x+y)*y*(2*x + y + 2*x*y)/2. - _Stefano Spezia_, Feb 21 2024
%e A361226 The rows are
%e A361226   0, 0,  1,  3,  6,  10,  15,  21, ...   = A161680
%e A361226   0, 1,  5, 12, 22,  35,  51,  70, ...   = A000326
%e A361226   0, 2,  9, 21, 38,  60,  87, 119, ...   = A005476
%e A361226   0, 3, 13, 30, 54,  85, 123, 168, ...   = A022264
%e A361226   0, 4, 17, 39, 70, 110, 159, 217, ...   = A022266
%e A361226   ... .
%e A361226 Columns: A000004, A001477, A016813, A017197=3*A016777, 2*A017101, 5*A016873, 3*A017581, 7*A017017, ... (coefficients from A026741).
%e A361226 Difference between two consecutive rows: A000290. Hence A143844.
%e A361226 This square array read by antidiagonals leads to the triangle
%e A361226   0
%e A361226   0   0
%e A361226   0   1   1
%e A361226   0   2   5   3
%e A361226   0   3   9  12   6
%e A361226   0   4  13  21  22  10
%e A361226   0   5  17  30  38  35  15
%e A361226   ... .
%t A361226 T[n_, k_] := k*((2*n + 1)*k - 1)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Mar 05 2023 *)
%o A361226 (PARI) a(n) = { my(row = (sqrtint(8*n+1)-1)\2, column = n - binomial(row + 1, 2)); binomial(column, 2) + column^2 * (row - column) } \\ _David A. Corneth_, Mar 05 2023
%o A361226 (Python) # Seen as a triangle:
%o A361226 from functools import cache
%o A361226 @cache
%o A361226 def Trow(n: int) -> list[int]:
%o A361226     if n == 0: return [0]
%o A361226     r = Trow(n - 1)
%o A361226     return [r[k] + k * k if k < n else r[n - 1] + n - 1 for k in range(n + 1)]
%o A361226 for n in range(7): print(Trow(n)) # _Peter Luschny_, Mar 05 2023
%Y A361226 Cf. A000004, A000290, A000326, A001477, A002414, A005476, A016777, A016813, A016873, A017017, A017101, A017197, A017581, A022264, A022266, A026741, A034827, A160378, A161680, A360962.
%Y A361226 Cf. A002492, A033275, A143844, A299120.
%K A361226 nonn,tabl
%O A361226 0,8
%A A361226 _Paul Curtz_, Mar 05 2023