OFFSET
0,1
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k) * StirlingS1(n, n-k).
Sum_{k=0..n} T(n, k) = 2^n + 2*342111(n) + Sum_{k=0..n} binomial(k*(n-k), n). - G. C. Greubel, Jun 03 2021
EXAMPLE
Triangle begins as:
4;
1, 1;
1, 4, 1;
1, 15, 15, 1;
1, 76, 249, 76, 1;
1, 485, 3516, 3516, 485, 1;
1, 3606, 46623, 101354, 46623, 3606, 1;
1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1;
1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1;
MATHEMATICA
T[n_, k_]:= Binomial[n, k] + Binomial[k*(n-k), n] + 2*(-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 03 2021 *)
PROG
(Magma)
A155826:= func< n, k | Binomial(n, k) + Binomial(k*(n-k), n) + 2*(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
[A155826(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 03 2021
(Sage)
def A155826(n, k): return binomial(n, k) + binomial(k*(n-k), n) + 2*stirling_number1(n, k)*stirling_number1(n, n-k)
flatten([[A155826(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 03 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 28 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 03 2021
STATUS
approved