A166036 -id:A166036 - cp's OEIS frontend

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

Paul D. Hanna, Sep 21 2005

Signed version of A108561. Row sums equal A084247. The n-th unsigned row sum = A001045(n) + 1 (Jacobsthal numbers). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms.
Equals row reversal of triangle A112468 up to sign, where A112468 is the Riordan array (1/(1-x),x/(1+x)). - Paul D. Hanna, Jan 20 2006
The elements here match A108561 in absolute value, but the signs are crucial to the properties that the matrix A112555 exhibits; the main property being T^m = I + m*(T - I). This property is not satisfied by A108561. - Paul D. Hanna, Nov 10 2009
Eigensequence of the triangle = A140165. - Gary W. Adamson, Jan 30 2009
Triangle T(n,k), read by rows, given by [1,-2,0,0,0,0,0,0,0,...] DELTA [1,0,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A112468 (reversed rows), A108561, A084247, A001045, A072547, A112556, A279006, A140165.
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)

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 *)

{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(""))

{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(""))

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

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)

A166149 a(n) = (5^n + 10*(-6)^n)/11.

Original entry on oeis.org

1, -5, 35, -185, 1235, -6785, 43835, -247385, 1562435, -8983985, 55857035, -325376585, 2001087635, -11762385185, 71795014235, -424666569785, 2578516996835, -15318514090385, 92674023995435, -552229446706985

Offset: 0

Philippe Deléham, Oct 08 2009

From Klaus Brockhaus, Oct 14 2009: (Start)
Fourth binomial transform of A014992.
Sixth binomial transform is A001020 preceded by 1.
Lim_{n -> infinity} a(n)/a(n-1) = -6. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (-1,30).

Cf. A166035, A166036.

Cf. A014992 (q-integers for q=-10), A001020 (powers of 11).

[(5^n+10*(-6)^n)/11: n in [0..30]]; // Vincenzo Librandi, May 02 2011

CoefficientList[Series[(1-4x)/(1+x-30x^2), {x,0,40}], x]  (* Harvey P. Dale, Mar 11 2011 *)
LinearRecurrence[{-1,30},{1,-5},20] (* Harvey P. Dale, Jan 20 2022 *)

a(n)=(5^n+10*(-6)^n)/11 \\ Charles R Greathouse IV, May 02 2016

a(n) = 30*a(n-2)-a(n-1), a(0)= 1, a(1)= -5.
G.f.: (1-4x)/(1+x-30*x^2).
a(n) = Sum_{k=0..n} A112555(n,k)*(-6)^k.
E.g.f.: (1/11)*(exp(5*x) + 10*exp(-6*x)). - G. C. Greubel, May 01 2016

A166152 a(n) = (6^n+12*(-7)^n)/13.

Original entry on oeis.org

1, -6, 48, -300, 2316, -14916, 112188, -738660, 5450556, -36474276, 265397628, -1797317220, 12944017596, -88431340836, 632080079868, -4346196394980, 30893559749436, -213433808338596, 1510963317814908

Offset: 0

Philippe Deléham, Oct 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 42).

Cf. A166035, A166036, A166149.

LinearRecurrence[{-1,42}, {1,-6}, 50] (* G. C. Greubel, May 01 2016 *)
Table[(6^n+12*(-7)^n)/13,{n,0,30}] (* Harvey P. Dale, Jun 01 2019 *)

a(n)=(6^n+12*(-7)^n)/13 \\ Charles R Greathouse IV, May 02 2016

a(n) = 42*a(n-2)-a(n-1), a(0)= 1, a(1)= -6, for n>1.
G.f.: (1-5x)/(1+x-42*x^2).
a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*(-7)^k.
E.g.f.: (1/13)*(exp(6*x) + 12*exp(-7*x)). - G. C. Greubel, May 01 2016

A166153 a(n) = (7^n+14*(-8)^n)/15.

Original entry on oeis.org

1, -7, 63, -455, 3983, -29463, 252511, -1902439, 16043055, -122579639, 1020990719, -7885450503, 65060930767, -506646158935, 4150058281887, -32522243182247, 264925506967919, -2086171125173751, 16921999515377215

Offset: 0

Philippe Deléham, Oct 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 56).

Cf. A112555, A166035, A166036, A166149, A166152.

LinearRecurrence[{-1, 56}, {1, -7}, 50] (* G. C. Greubel, May 01 2016 *)

a(n) = 56*a(n-2)-a(n-1), a(0)= 1, a(1)= -7, for n>1.
G.f.: (1-6*x)/(1+x-56*x^2).
a(n) = Sum_{k=0..n} A112555(n,k)*(-8)^k.
E.g.f.: (1/15)*(exp(7*x) + 14*exp(-8*x)). - G. C. Greubel, May 01 2016

A166157 a(n) = (8^n+16*(-9)^n)/17.

Original entry on oeis.org

1, -8, 80, -656, 6416, -53648, 515600, -4378256, 41501456, -356735888, 3344840720, -29029824656, 269858356496, -2360005731728, 21789807399440, -191710220083856, 1760576352843536, -15563712198881168

Offset: 0

Philippe Deléham, Oct 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 72).

Cf. A166035, A166036, A166149, A166152, A166153.

LinearRecurrence[{-1,72},{1,-8},20] (* Harvey P. Dale, Jun 23 2012 *)

a(n)=(8^n+16*(-9)^n)/17 \\ Charles R Greathouse IV, May 02 2016

a(n) = 72*a(n-2)-a(n-1), a(0)= 1, a(1)= -8, for n>1.
G.f.: (1-7x)/(1+x-72*x^2).
a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*(-9)^k.
E.g.f.: (1/17)*(exp(8*x) + 16*exp(-9*x)). - G. C. Greubel, May 01 2016

cp's OEIS Frontend

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.

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A166149 a(n) = (5^n + 10*(-6)^n)/11.

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A166152 a(n) = (6^n+12*(-7)^n)/13.

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A166153 a(n) = (7^n+14*(-8)^n)/15.

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A166157 a(n) = (8^n+16*(-9)^n)/17.

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