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

0,9

Comments

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

Links

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

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

Example

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; ...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A099793 A273895 A363044 * A294118 A343648 A359363

Adjacent sequences: A086326 A086327 A086328 * A086330 A086331 A086332

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Aug 30 2003, Jun 12 2007

Status

approved