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A054090 Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T.

%I A054090 #16 Jul 19 2024 03:13:03
%S A054090 1,1,1,1,2,1,1,4,2,3,1,10,6,8,7,1,32,22,26,24,25,1,130,98,108,104,106,
%T A054090 105,1,652,522,554,544,548,546,547,1,3914,3262,3392,3360,3370,3366,
%U A054090 3368,3367,1,27400,23486,24138,24008,24040,24030,24034,24032,24033
%N A054090 Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T.
%H A054090 G. C. Greubel, <a href="/A054090/b054090.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A054090 T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j).
%F A054090 Sum_{k=0..n} T(n, k) = A054091(n).
%e A054090 Triangle begins as:
%e A054090   1;
%e A054090   1,     1;
%e A054090   1,     2,     1;
%e A054090   1,     4,     2,     3;
%e A054090   1,    10,     6,     8,     7;
%e A054090   1,    32,    22,    26,    24,    25;
%e A054090   1,   130,    98,   108,   104,   106,   105;
%e A054090   1,   652,   522,   554,   544,   548,   546,   547;
%e A054090   1,  3914,  3262,  3392,  3360,  3370,  3366,  3368,  3367;
%e A054090   1, 27400, 23486, 24138, 24008, 24040, 24030, 24034, 24032, 24033;
%t A054090 T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, Sum[T[n-1,j], {j,0,n-1}], T[n,k-1] - (-1)^k*Sum[T[n-k,j], {j,0,n-k}]]];
%t A054090 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 23 2022 *)
%o A054090 (PARI) {T(n, k)= local(A); if(k<0||k>n, 0, if(k==0, 1, A=vector(n, i, (i>1)+1); for(i=2, n-1, A[i+1]=(i-1)*A[i]+2); sum(i=0, k-1, (-1)^i*A[n-i])))} /* _Michael Somos_, Nov 19 2006 */
%o A054090 (SageMath)
%o A054090 @CachedFunction
%o A054090 def T(n, k): # T = A054090
%o A054090     if (k==0): return 1
%o A054090     elif (k==1): return sum(T(n-1, j) for j in (0..n-1))
%o A054090     else: return T(n, k-1) - (-1)^k*sum(T(n-k, j) for j in (0..n-k))
%o A054090 flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 23 2022
%Y A054090 Cf. A054091 (row sums).
%K A054090 nonn,tabl,eigen
%O A054090 0,5
%A A054090 _Clark Kimberling_