OFFSET
1,5
COMMENTS
T(n,k) is the number of blades of dimension (n-k) in the canonical basis of graduated blades (see Early link).
T(n+1,k+1) is the number of ways to partition n labeled balls into a sack containing k nonempty necklaces of nonempty bags; balls can be placed in the sack directly, or grouped into bags which are hung upon a necklace. - Natalia L. Skirrow, Dec 21 2025
LINKS
G. C. Greubel, Rows n=1..100 of triangle, flattened
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
FORMULA
From Natalia L. Skirrow, Dec 22 2025: (Start)
E.g.f.: y/(y-1)*((2-e^x)^(1-y)-1), k-th column: 1-(2-e^x)*Sum_{i=0..k-1} log(1/(2-e^x))^i/i!.
E.g.f. (with offset 0 in both variables): e^x/(2-e^x)^y, k-th column: e^x*log(1/(2-e^x))^k/k!. (End)
EXAMPLE
Triangle begins:
1,
1, 1,
1, 4, 1,
1, 15, 9, 1,
1, 66, 66, 16, 1,
1, 365, 500, 190, 25, 1,
...
MATHEMATICA
T[n_, k_]:= Sum[StirlingS2[n, j]*Abs[StirlingS1[j-1, k-1]], {j, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Oct 14 2018 *)
MatrixForm[Transpose[Table[CoefficientList[Series[1-(2-E^x)*Sum[(-1)^i*Log[2-E^x]^i/i!, {i, 0, k-1}], {x, 0, 8}], x]*Range[0, 8]!, {k, 1, 8}]]] (* Natalia L. Skirrow, Dec 22 2025 *)
PROG
(PARI) T(n, k) = sum(i=1, n, stirling(n, i, 2)*abs(stirling(i-1, k-1, 1)));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print);
(Magma) [[(&+[StirlingSecond(n, i)*Abs(StirlingFirst(i-1, k-1)): i in [1..n]]): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Oct 14 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, Oct 09 2018
STATUS
approved