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.

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, 105, 1, 652, 522, 554, 544, 548, 546, 547, 1, 3914, 3262, 3392, 3360, 3370, 3366, 3368, 3367, 1, 27400, 23486, 24138, 24008, 24040, 24030, 24034, 24032, 24033

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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).

Sum_{k=0..n} T(n, k) = A054091(n).

Example

Triangle begins as:
  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,   105;
  1,   652,   522,   554,   544,   548,   546,   547;
  1,  3914,  3262,  3392,  3360,  3370,  3366,  3368,  3367;
  1, 27400, 23486, 24138, 24008, 24040, 24030, 24034, 24032, 24033;

Mathematica

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}]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 23 2022 *)

Prog

(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 */
(SageMath)
@CachedFunction
def T(n, k): # T = A054090
    if (k==0): return 1
    elif (k==1): return sum(T(n-1, j) for j in (0..n-1))
    else: return T(n, k-1) - (-1)^k*sum(T(n-k, j) for j in (0..n-k))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 23 2022

Crossrefs

Cf. A054091 (row sums).

Sequence in context: A064191 A127420 A129033 * A239456 A122517 A256098

Adjacent sequences: A054087 A054088 A054089 * A054091 A054092 A054093

Keyword

nonn,tabl,eigen

Author

Clark Kimberling

Status

approved