A124038 - OEIS

%I #15 Jan 23 2025 00:17:01

%S 1,-2,1,-1,-2,1,2,-2,-2,1,1,4,-3,-2,1,-2,3,6,-4,-2,1,-1,-6,6,8,-5,-2,

%T 1,2,-4,-12,10,10,-6,-2,1,1,8,-10,-20,15,12,-7,-2,1,-2,5,20,-20,-30,

%U 21,14,-8,-2,1,-1,-10,15,40,-35,-42,28,16,-9,-2,1

%N Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

%H G. C. Greubel, <a href="/A124038/b124038.txt">Rows n = 0..50 of the traingle, flattened</a>

%F From _G. C. Greubel_, Jan 22 2025: (Start)

%F T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

%F T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).

%F T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

%e Triangular sequence begins as:

%e    1;

%e   -2,   1;

%e   -1,  -2,   1;

%e    2,  -2,  -2,   1;

%e    1,   4,  -3,  -2,   1;

%e   -2,   3,   6,  -4,  -2,   1;

%e   -1,  -6,   6,   8,  -5,  -2,  1;

%e    2,  -4, -12,  10,  10,  -6, -2,  1;

%e    1,   8, -10, -20,  15,  12, -7, -2,  1;

%e   -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;

%e   -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

%t (* First program *)

%t t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];

%t M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];

%t Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten

%t (* Second program *)

%t T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 22 2025 *)

%o (SageMath)

%o @CachedFunction

%o def A124038(n,k):

%o     if n< 0: return 0

%o     if n==0: return 1 if k == 0 else 0

%o     h = 2*A124038(n-1,k) if n==1 else 0

%o     return A124038(n-1,k-1) - A124038(n-2,k) - h

%o for n in (0..9): [A124038(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012

%o (SageMath)

%o from sage.combinat.q_analogues import q_stirling_number2

%o def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)

%o print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 22 2025

%o (Magma)

%o function T(n,k) // T = A124038

%o   if k lt 0 or k gt n then return 0;

%o   elif k ge n-2 then return k-n + (-1)^(n+k);

%o   else return T(n-1,k-1) - T(n-2,k);

%o   end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 22 2025

%Y Cf. A005809, A045721, A066170.

%Y Row reversal of: A374439.

%Y Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).

%Y Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).

%Y Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

%K sign

%O 0,2

%A _Gary W. Adamson_ and _Roger L. Bagula_, Nov 03 2006

%E Edited by _G. C. Greubel_, Jan 22 2025