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A123158 Square array related to Bell numbers read by antidiagonals.

%I A123158 #22 Jul 19 2023 07:56:58
%S A123158 1,1,1,2,2,1,5,5,3,1,15,15,10,5,1,52,52,37,22,6,1,203,203,151,99,31,9,
%T A123158 1,877,877,674,471,160,61,10,1,4140,4140,3263,2386,856,385,75,14,1,
%U A123158 21147,21147,17007,12867,4802,2416,520,135,15,1
%N A123158 Square array related to Bell numbers read by antidiagonals.
%H A123158 G. C. Greubel, <a href="/A123158/b123158.txt">Antidiagonals n = 0..50, flattened</a>
%F A123158 A(n, k) = 0 if n < 0, A(0, k) = 1 for k >= 0, A(n, k) = A(n, k-1) + (1/2)*(k+1)*A(n-1, k+1) if k is an odd number, A(n, k) = A(n, k-1) + A(n-1, k+1) if k is an even number (array).
%F A123158 A(n, 0) = A000110(n).
%F A123158 A(n, 1) = A000110(n+1).
%F A123158 A(n, 2) = A005493(n).
%F A123158 A(n, 3) = A033452(n).
%F A123158 A(n, 4) = A003128(n+2).
%F A123158 T(n, k) = A(n-k, k) (antidiagonals).
%e A123158 Square array, A(n, k), begins:
%e A123158    1,   1,   1,    1,    1, ... (Row n=0: A000012);
%e A123158    1,   2,   3,    5,    6, ... (Row n=1: A117142);
%e A123158    2,   5,  10,   22,   31, ...;
%e A123158    5,  15,  37,   99,  160, ...;
%e A123158   15,  52, 151,  471,  856, ...;
%e A123158   52, 203, 674, 2386, 4802, ...;
%e A123158 Antidiagonals, T(n, k), begin as:
%e A123158     1;
%e A123158     1,   1;
%e A123158     2,   2,   1;
%e A123158     5,   5,   3,   1;
%e A123158    15,  15,  10,   5,   1;
%e A123158    52,  52,  37,  22,   6,  1;
%e A123158   203, 203, 151,  99,  31,  9,   1;
%e A123158   877, 877, 674, 471, 160, 61,  10,  1;
%t A123158 A[0, _?NonNegative] = 1;
%t A123158 A[n_, k_]:= A[n, k]= If[n<0 || k<0, 0, If[OddQ[k], A[n, k-1] + (1/2)(k+1) A[n-1, k+1], A[n, k-1] + A[n-1, k+1]]];
%t A123158 Table[A[n-k, k], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Feb 21 2020 *)
%o A123158 (Magma)
%o A123158 function A(n,k)
%o A123158   if k lt 0 or n lt 0 then return 0;
%o A123158   elif n eq 0 then return 1;
%o A123158   elif (k mod 2) eq 1 then return A(n,k-1) + (1/2)*(k+1)*A(n-1,k+1);
%o A123158   else return A(n,k-1) + A(n-1,k+1);
%o A123158   end if;
%o A123158 end function;
%o A123158 T:= func< n,k | A(n-k,k) >;
%o A123158 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 18 2023
%o A123158 (SageMath)
%o A123158 @CachedFunction
%o A123158 def A(n,k):
%o A123158     if (k<0 or n<0): return 0
%o A123158     elif (n==0): return 1
%o A123158     elif (k%2==1): return A(n,k-1) +(1/2)*(k+1)*A(n-1,k+1)
%o A123158     else: return A(n,k-1) +A(n-1,k+1)
%o A123158 def T(n,k): return A(n-k,k)
%o A123158 flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Jul 18 2023
%Y A123158 Columns include: A000110 (Bell numbers), A003128, A005493, A033452.
%Y A123158 Rows include: A000012, A117142.
%K A123158 easy,nonn,tabl
%O A123158 0,4
%A A123158 _Philippe Deléham_, Oct 01 2006