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 A133607 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = -1,  with 0 <= k <= n. 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS Previous name: Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. LINKS 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 EXAMPLE 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;   ... 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 *) PROG (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 CROSSREFS Another version is A108299. Unsigned version is A103631 (T(n,k) = A103631(n,k)*A057077(k)). Sequence in context: A131255 A198295 A221857 * A103631 A263191 A192517 Adjacent sequences:  A133604 A133605 A133606 * A133608 A133609 A133610 KEYWORD sign,tabl AUTHOR Philippe Deléham, Dec 27 2007 EXTENSIONS New name from Peter Luschny, Mar 09 2020 STATUS approved

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Last modified January 21 02:20 EST 2021. Contains 340332 sequences. (Running on oeis4.)