A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.
1, 1, 1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -2, 0, 1, 1, 3, 4, 2, 1, 1, -1, -4, -7, -6, -3, 0, 1, 1, 5, 11, 13, 9, 3, 1, 1, -1, -6, -16, -24, -22, -12, -4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, -1, -10, -46, -128, -239, -314, -296, -200, -95, -30, -6, 0
Offset: 0
Examples
Triangle T begins: 1; 1, 1; -1, 0, 1; 1, 1, 1, 1; -1, -2, -2, 0, 1; 1, 3, 4, 2, 1, 1; -1, -4, -7, -6, -3, 0, 1; 1, 5, 11, 13, 9, 3, 1, 1; -1, -6, -16, -24, -22, -12, -4, 0, 1; 1, 7, 22, 40, 46, 34, 16, 4, 1, 1; -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1; ... Matrix log, log(T) = T - I, begins: 0; 1, 0; -1, 0, 0; 1, 1, 1, 0; -1, -2, -2, 0, 0; 1, 3, 4, 2, 1, 0; -1, -4, -7, -6, -3, 0, 0; ... Matrix inverse, T^-1 = 2*I - T, begins: 1; -1, 1; 1, 0, 1; -1, -1, -1, 1; 1, 2, 2, 0, 1; -1, -3, -4, -2, -1, 1; ... where adjacent sums in row n of T^-1 gives row n+1 of T.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1080
Crossrefs
From Philippe Deléham, Oct 07 2009: (Start)
Sum_{k=0..n} T(n, k)*x^(n-k) = A165760(n), A165759(n), A165758(n), A165755(n), A165752(n), A165746(n), A165751(n), A165747(n), A000007(n), A000012(n), A084247(n), A165553(n), A165622(n), A165625(n), A165638(n), A165639(n), A165748(n), A165749(n), A165750(n) for x= -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} T(n, k)*x^k = A166157(n), A166153(n), A166152(n), A166149(n), A166036(n), A166035(n), A091004(n+1), A077925(n), A000007(n), A165326(n), A084247(n), A165405(n), A165458(n), A165470(n), A165491(n), A165505(n), A165506(n), A165510(n), A165511(n) for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. (End)
Programs
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Mathematica
Clear[t]; t[0, 0] = 1; t[n_, 0] = (-1)^(Mod[n, 2]+1); t[n_, n_] = 1; t[n_, k_] /; k == n-1 := t[n, k] = Mod[n, 2]; t[n_, k_] /; 0 < k < n-1 := t[n, k] = -t[n-1, k] - t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *) -
PARI
{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff( polcoeff( (1+2*x+x*y)/((1-x*y)*(1+x+x*y)),n,X),k,Y)} for(n=0,12, for(k=0,n, print1(T(n,k),", "));print("")) -
PARI
{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)),n,X),k,Y)} for(n=0,12, for(k=0,n, print1(T(n,k),", "));print("")) -
Sage
def A112555_row(n): @cached_function def prec(n, k): if k==n: return 1 if k==0: return 0 return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1)) return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n+1)] for n in (0..12): print(A112555_row(n)) # Peter Luschny, Mar 16 2016
Formula
G.f.: 1/(1-x*y) + x/((1-x*y)*(1+x+x*y)).
The m-th matrix power T^m has the g.f.: 1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)).
Recurrence: T(n, k) = [T^-1](n-1, k) + [T^-1](n-1, k-1), where T^-1 is the matrix inverse of T.
From Peter Bala, Jun 23 2025: (Start)
T^z = exp(z*log(T)) = I + z*(T - I) for arbitrary complex z, where I is the identity array.
exp(T) = e*T. More generally, exp(z * T^u) = exp(z)*T^(u*z) = exp(z)*I + u*z*exp(z)*(T - I).
sin(z * T^u) = sin(z)*I + u*z*cos(z)*(T - I).
cos(z * T^u) = cos(z)*I - u*z*sin(z)*(T - I).
tan(z * T^u) = tan(z)*I + u*z*sec(z)^2*(T - I).
Chebyshev_T(n, T^u) = I + (n^2)*u*(T - I) and
Legendre_P(n, T^u) = I + (n*(n+1)/2)*u*(T - I).
More generally, for n >= 1,
Chebyshev_T(n, z*T^u) = Chebyshev_T(n, z)*I + n*u*z*Chebyshev_U(n-1, z)*(T - I) and
Legendre_P(n, z*T^u) = Legendre_P(n, z)*I + u*Q(n, z)*(T - I), where Q(1, z) = z and Q(n, z) = n*Legendre_P(n, z) + Q(n-1, z)/z for n > 1.
All the above properties may also hold for the triangle A279006. (End)
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