A133607 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = -1, with 0 <= k <= n.
1, 0, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, -2, 1, 0, 1, -1, -3, 2, 1, 0, 1, -1, -4, 3, 3, -1, 0, 1, -1, -5, 4, 6, -3, -1, 0, 1, -1, -6, 5, 10, -6, -4, 1, 0, 1, -1, -7, 6, 15, -10, -10, 4, 1, 0, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 0, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1
Offset: 0
Examples
Triangle begins: 1; 0, 1; 0, 1, -1; 0, 1, -1, -1; 0, 1, -1, -2, 1; 0, 1, -1, -3, 2, 1; 0, 1, -1, -4, 3, 3, -1; 0, 1, -1, -5, 4, 6, -3, -1; 0, 1, -1, -6, 5, 10, -6, -4, 1; 0, 1, -1, -7, 6, 15, -10, -10, 4, 1; 0, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1; 0, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1; 0, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1; ... Triangle A103631 begins: 1; 0, 1; 0, 1, 1; 0, 1, 1, 1; 0, 1, 1, 2, 1; 0, 1, 1, 3, 2, 1; 0, 1, 1, 4, 3, 3, 1; 0, 1, 1, 5, 4, 6, 3, 1; 0, 1, 1, 6, 5, 10, 6, 4, 1; 0, 1, 1, 7, 6, 15, 10, 10, 4, 1; 0, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1; 0, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1; 0, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1; ... Triangle A108299 begins: 1; 1, -1; 1, -1, -1; 1, -1, -2, 1; 1, -1, -3, 2, 1; 1, -1, -4, 3, 3, -1; 1, -1, -5, 4, 6, -3, -1; 1, -1, -6, 5, 10, -6, -4, 1; 1, -1, -7, 6, 15, -10, -10, 4, 1; 1, -1, -8, 7, 21, -15, -20, 10, 5, -1; 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1; 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1; ...
Crossrefs
Programs
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Mathematica
m = 13 (* DELTA is defined in A084938 *) DELTA[Join[{0, 1}, Table[0, {m}]], Join[{1, -2, 1}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *) 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, k, -1], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2020 *)
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Sage
from sage.combinat.q_analogues import q_stirling_number2 for n in (0..9): print([q_stirling_number2(n,k).substitute(q=-1) for k in [0..n]]) # Peter Luschny, Mar 09 2020
Formula
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A057077(n), A010892(n), A000012(n), A001519(n), A001835(n), A004253(n), A001653(n), A049685(n-1), A070997(n-1), A070998(n-1), A072256(n), A078922(n), A077417(n-1), A085260(n), A001570(n-1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 respectively .
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A010892(n), A133631(n), A133665(n), A133666(n), A133667(n), A133668(n), A133669(n), A133671(n), A133672(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively .
G.f.: (1-x+y*x)/(1-x+y^2*x^2). - Philippe Deléham, Mar 14 2012
T(n,k) = T(n-1,k) - T(n-2,k-2), T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = -1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 14 2012
Extensions
New name from Peter Luschny, Mar 09 2020
Comments