A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 11, 1, 0, 1, 16, 48, 26, 1, 0, 1, 25, 140, 202, 57, 1, 0, 1, 36, 325, 916, 747, 120, 1, 0, 1, 49, 651, 3045, 5071, 2559, 247, 1, 0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1, 0, 1, 81, 1968, 19404, 84456, 159736, 117962, 26520, 1013, 1

Offset: 0

Author

Philippe Deléham, Aug 30 2003, Jun 12 2007

Keywords

• easy
• nonn
• tabl

Comments

See A087903 for another version (transposed). - Philippe Deléham, Jun 13 2004

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  4,    1;
  0, 1,  9,   11,    1;
  0, 1, 16,   48,   26,     1;
  0, 1, 25,  140,  202,    57,     1;
  0, 1, 36,  325,  916,   747,   120,    1;
  0, 1, 49,  651, 3045,  5071,  2559,  247,   1;
  0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1; ...
		

Links

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

Crossrefs

Cf. A000290, A000295, A074664, A084938, A086211, A171151.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[n==0, 1, StirlingS2[n, k] + Sum[(k-m-1)*T[n-j-1, k- m]*StirlingS2[j, m], {m,0,k-1}, {j,0,n-2}]];
A086329[n_, k_]:= T[n,n-k+1];
Table[A086329[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 21 2022 *)

SageMath

@CachedFunction
def T(n,k): # T=A087903
    if (n==0): return 1
    else: return stirling_number2(n, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
def A086329(n,k): return T(n, n-k+1)
flatten([[A086329(n, k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Jun 21 2022

Formula

Sum_{k=0..n} T(n, k) = A086211(n, 0).

T(n, 1) = 1, n > 0.

T(n, 2) = (n-1)^2, n > 0.

T(k+1, k) = 2^(k+1) - k - 2 = A000295(k+1).

Sum_{k=0..n} T(n, k) = A074664(n+1). - Philippe Deléham, Jun 13 2004

Sum_{k=0..n} T(n,k)*2^k = A171151(n). - Philippe Deléham, Dec 05 2009

T(n, k) = A087903(n, n-k+1). - G. C. Greubel, Jun 21 2022