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
KEYWORD
AUTHOR
Philippe Deléham, Aug 30 2003, Jun 12 2007
STATUS
approved