A124040 Triangle read by rows: characteristic polynomials of certain matrices, see Mathematica program.
3, 3, -1, 8, -6, 1, 20, -24, 9, -1, 45, -84, 50, -12, 1, 125, -275, 225, -85, 15, -1, 320, -864, 900, -468, 129, -18, 1, 845, -2639, 3339, -2219, 840, -182, 21, -1, 2205, -7896, 11756, -9528, 4610, -1368, 244, -24, 1, 5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1, 15125, -67650, 130975, -144660, 101065, -46746, 14525, -3000, 395, -30, 1
Offset: 1
Examples
Triangle begins:
3;
3, -1;
8, -6, 1;
20, -24, 9, -1;
45, -84, 50, -12, 1;
125, -275, 225, -85, 15, -1;
320, -864, 900, -468, 129, -18, 1;
845, -2639, 3339, -2219, 840, -182, 21, -1;
2205, -7896, 11756, -9528, 4610, -1368, 244, -24, 1;
5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Programs
-
Mathematica
T[n_, m_, d_]:= If[n==m, 3, If[n==m-1 || n==m+1, 1, If[(n==1 && m==d) || (n==d && m==1), 1, 0]]]; M[d_]:= Table[T[n, m, d], {n,d}, {m,d}]; Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,12}]]
Formula
From G. C. Greubel, Feb 03 2025: (Start)
T(n, 1) = A099921(n-1) + 3*[n=1] - 2*[n=2] + 3*[n=3].
T(n, 2) = -(n-1)*Fibonacci(2*n-2).
T(n, 3) = (1/10)*(n-1)*(2*(n-1)*Fibonacci(2*n-1) - (n+2)*Fibonacci(2*n-2)).
T(n, 4) = (1/150)*(n-1)*(18*(n-1)*Fibonacci(2*n-1) - (5*n^2 - n + 18)*Fibonacci(2*n-2)).
T(n, 5) = (1/600)*(n-1)*(2*(n-1)*(n^2-2*n+24)*Fibonacci(2*n-1) - (n^3+15*n^2 -10*n+48)*Fibonacci(2*n-2)).
T(n, n) = (-1)^(n-1) + 2*[n=1].
T(n, n-1) = 3*(-1)^n*(n-1).
T(n, n-2) = (1/2)*(-1)^(n+1)*(n-1)*(9*n-20) + [n=3].
T(n, n-3) = (3/2)*(-1)^n*(n-1)*(n-3)*(3*n-8) + 2*[n=4].
T(n, n-4) = (1/8)*(-1)^(n-1)*n*(n-3)*(27*n^2-117*n+130) - 2*[n=5].
T(n, n-5) = (3/40)*(-1)^n*(n-1)*(n-4)*(n-5)*(27*n^2-195*n+362) + 2*[n=6].
T(n, n-6) = (1/240)*(-1)^(n-1)*(n-1)*(n-5)*(n-6)(243*n^3-2997*n^2+12528*n -17752) -2*[n=7].
T(2*n-1, n) = 2*(-1)^(n-1)*A370280(n-1) + [n=1].
Sum_{k=1..n} T(n, k) = A010675(n-1) + 3*[n=1] -2*[n=2] +3*[n=3].
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A201630(n-2) - A201630(n-1) + (1/4)*[n=1] + (7/2)*[n=2] + 2*[n=3].
(End)
Comments