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A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

%I A261720 #26 Aug 16 2024 08:36:18
%S A261720 1,1,4,1,5,10,1,6,14,20,1,7,18,30,35,1,8,22,40,55,56,1,9,26,50,75,91,
%T A261720 84,1,10,30,60,95,126,140,120,1,11,34,70,115,161,196,204,165,1,12,38,
%U A261720 80,135,196,252,288,285,220,1,13,42,90,155,231,308,372,405,385,286
%N A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.
%C A261720 First few sequences in the array:
%C A261720   1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
%C A261720   1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
%C A261720   1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
%C A261720   1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
%C A261720   1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
%C A261720   1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
%C A261720   1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
%C A261720   ...
%C A261720 The corresponding bases to rows are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ...
%D A261720 Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 194.
%F A261720 T(n,k) = A080851(n,k).
%F A261720 Given: first sequence in the array is A000292: (1, 4, 10, 20, 35, ...) Subsequent rows are generated by adding (0, 1, 4, 10, 20, 35, ...) to the current row.
%F A261720 n-th row is the binomial transform of row 3 in Pascal's triangle (1,n) followed by zeros.  Alternatively, begin with (1, 4, 10, 20, ...) being the binomial transform of (1, 3, 3, 1, 0, 0, 0, ...). Add (0, 1, 2, 1, 0, 0, 0, ...) to the latter to obtain the inverse binomial transform of the next row: (1, 5, 14, 30, 55,..); then repeat the operation.
%F A261720 The row starting (1, N, ...) is the 3rd partial sum of (1, (N-3), (N-3), (N-3), ...).
%F A261720 From _Stefano Spezia_, Aug 15 2024: (Start)
%F A261720 T(n,k) = k*(k + 1)*((k - 1)*n + 3)/6.
%F A261720 G.f. as array: x*y*(1 + x*(y - 1))/((1 - x)^2*(1 - y)^4).
%F A261720 E.g.f. as array: exp(y)*y*(exp(x)*(6 + 3*(1 + x)*y + x*y^2) - 3*(2 + y))/6. (End)
%e A261720 Row 2: (1, 5, 14, 30, 55, ...) = (1, 4, 10, 20, 35, ...) + (0, 1, 4, 10, 20, 35, ...).
%e A261720 (1, 7, 22, 50, ...) is the binomial transform of (1, 6, 9, 4, 0, 0, 0, ...) 3rd row in Pascal's triangle (1,4) followed by zeros. (1, 7, 22, 50, ...) is the third partial sum of (1, 4, 4, 4, ...).
%t A261720 T[n_,k_]:=k(k+1)((k-1)n+3)/6; Flatten[Table[T[n-k+1,k],{n,11},{k,n}]] (* _Stefano Spezia_, Aug 15 2024 *)
%Y A261720 Cf. A000292, A000330, A002411, A002412, A002413, A002414, A007584, A220084.
%Y A261720 Similar to A080851 but without row n=0.
%K A261720 nonn,tabl,easy
%O A261720 1,3
%A A261720 _Gary W. Adamson_, Aug 29 2015