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