A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 21, 18, 1, 1, 85, 255, 58, 1, 1, 341, 3400, 2575, 179, 1, 1, 1365, 44541, 106400, 24234, 543, 1, 1, 5461, 580398, 4300541, 3038714, 221886, 1636, 1, 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Formula

qStirling2(n, k, q) = qStirling2(n-1, k-1, q) + qBrackets(k+1, q)*qStirling2(n-1, k, q) with boundary values 0^k if n = 0 and n^0 if k = 0.

Note that also a second definition is used in the literature which has an additional factor q^k attached to the first term in the equation above. The two versions differ by a factor of q^binomial(k,2).

Example

[0] 1
[1] 1, 1
[2] 1, 5,     1
[3] 1, 21,    18,      1
[4] 1, 85,    255,     58,        1
[5] 1, 341,   3400,    2575,      179,       1
[6] 1, 1365,  44541,   106400,    24234,     543,      1
[7] 1, 5461,  580398,  4300541,   3038714,   221886,   1636,    1
[8] 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

Maple

qStirling2 := proc(n, k, q) option remember; with(QDifferenceEquations):
if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
qStirling2(n-1, k-1, p) + QBrackets(k+1, p)*qStirling2(n-1, k, p);
subs(p = q, expand(%)) end:
seq(seq(qStirling2(n, k, 3), k=0..n), n=0..9);

Mathematica

qStirling2[n_, k_, q_] /; 1 <= k <= n := (* q^(k-1) *) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[_, _, _] = 0;
Table[qStirling2[n + 1, k + 1, 3], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2020 *)

Crossrefs

T(n, 1) = A002450(n), T(n, n-1) = A000340(n).

Cf. A139382 (q=2), A333142.

Sequence in context: A176242 A036969 A080249 * A157154 A022168 A359993

Adjacent sequences: A333140 A333141 A333142 * A333144 A333145 A333146

Keyword

nonn,tabl

Author

Peter Luschny, Mar 09 2020

Status

approved