A320280 - OEIS

%I #15 Sep 08 2022 08:46:23

%S 1,1,1,1,4,1,1,15,9,1,1,66,66,16,1,1,365,500,190,25,1,1,2528,4215,

%T 2150,435,36,1,1,21259,40355,25235,6825,861,49,1,1,210430,438256,

%U 317632,105910,17836,1540,64,1,1,2393769,5352534,4338264,1693734,352926,40656,2556,81,1

%N Triangle T(n, k) = Sum_{i=1..n} Stirling2(n,i) * abs(Stirling1(i-1,k-1)), n >= 1, 1 <= k <= n.

%C T(n,k) is the number of blades of dimension (n-k) in the canonical basis of graduated blades (see Early link).

%H G. C. Greubel, <a href="/A320280/b320280.txt">Rows n=1..100 of triangle, flattened</a>

%H Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.

%e Triangle begins:

%e   1,

%e   1,   1,

%e   1,   4,   1,

%e   1,  15,   9,   1,

%e   1,  66,  66,  16,  1,

%e   1, 365, 500, 190, 25, 1,

%e   ...

%t 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 *)

%o (PARI) T(n, k) = sum(i=1, n, stirling(n,i,2)*abs(stirling(i-1,k-1,1)));

%o tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print);

%o (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

%Y Cf. A008275 (Stirling1), A008277 (Stirling2).

%K nonn,tabl

%O 1,5

%A _Michel Marcus_, Oct 09 2018