A091004 -id:A091004 - 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)

A166035 a(n) = (3^n+6*(-4)^n)/7.

Original entry on oeis.org

1, -3, 15, -51, 231, -843, 3615, -13731, 57111, -221883, 907215, -3569811, 14456391, -57294123, 230770815, -918300291, 3687550071, -14707153563, 58957754415, -235443597171, 942936650151, -3768259816203, 15083499618015

Offset: 0

Philippe Deléham, Oct 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,12).

Cf. A077925, A091004.

LinearRecurrence[{-1, 12}, {1, -3}, 100] (* G. C. Greubel, Apr 24 2016 *)

a(n)= (3^n+6*(-4)^n)/7;
for(n=0,33,print1(a(n),", "));

a(n) = -a(n-1) + 12*a(n-2), a(0) = 1, a(1) = -3, for n>1.
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-4)^k.
G.f.: (1-2x)/(1+x-12x^2).
E.g.f.: (1/7)*(exp(3*x) + 6*exp(-4*x)). - G. C. Greubel, Apr 24 2016

a(5) corrected by Tilman Neumann, Dec 31 2010

A166036 a(n) = (4^n+8*(-5)^n)/9.

Original entry on oeis.org

1, -4, 24, -104, 584, -2664, 14344, -67624, 354504, -1706984, 8797064, -42936744, 218878024, -1077612904, 5455173384, -27007431464, 136110899144, -676259528424, 3398477511304, -16923668079784

Offset: 0

Philippe Deléham, Oct 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,20).

Cf. A077925, A091004, A166035.

Table[(4^n+8(-5)^n)/9,{n,0,30}] (* or *) LinearRecurrence[{-1,20},{1,-4},30] (* Harvey P. Dale, Mar 05 2016 *)

vector(100, n, n--; (4^n+8*(-5)^n)/9) \\ Altug Alkan, Apr 24 2016

a(n) = -a(n-1) + 20*a(n-2), a(0) = 1, a(1) = -4, for n>1.
a(n) = Sum_{k=0..n} A112555(n,k)*(-5)^k.
G.f.: (1-3x)/(1+x-20x^2).
E.g.f.: (1/9)*(exp(4*x) + 8*exp(-5*x)). - G. C. Greubel, Apr 24 2016

A091003 Expansion of (1-3x^2)/((1-2x)(1+3x)).

Original entry on oeis.org

1, -1, 4, -10, 34, -94, 298, -862, 2650, -7822, 23722, -70654, 212986, -636910, 1914826, -5736286, 17225242, -51642958, 154994410, -464852158, 1394818618, -4183931566, 12552843274, -37656432670, 112973492314, -338912088334, 1016753042218

Offset: 0

Paul Barry, Dec 13 2003

Inverse binomial transform of A091000.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,6).

Concatenation([1], List([1..30], n -> (2^n + 4*(-3)^n)/10)); # G. C. Greubel, Feb 01 2019

[1] cat [(2^n + 4*(-3)^n)/10: n in [1..30]]; // G. C. Greubel, Feb 01 2019

CoefficientList[Series[(1-3x^2)/((1-2x)(1+3x)),{x,0,30}],x] (* Harvey P. Dale, Dec 23 2014 *)
Join[{1}, LinearRecurrence[{-1,6}, {-1,4}, 30]] (* G. C. Greubel, Feb 01 2019 *)

vector(30, n, n--; (2^n + 4*(-3)^n + 5*0^n)/10) \\ G. C. Greubel, Feb 01 2019

[1] + [(2^n + 4*(-3)^n)/10 for n in (1..30)] # G. C. Greubel, Feb 01 2019

2^n = A091003(n) + 3*A091004(n) + 6*A091005(n).
a(n) = (2^n + 4*(-3)^n + 5*0^n)/10.
E.g.f.: (exp(2*x) + 4*exp(-3*x) + 5)/10. - G. C. Greubel, Feb 01 2019

A091005 Expansion of x^2/((1-2x)(1+3*x)).

Original entry on oeis.org

0, 0, 1, -1, 7, -13, 55, -133, 463, -1261, 4039, -11605, 35839, -105469, 320503, -953317, 2876335, -8596237, 25854247, -77431669, 232557151, -697147165, 2092490071, -6275373061, 18830313487, -56482551853, 169464432775, -508359743893, 1525146340543

Offset: 0

Paul Barry, Dec 13 2003

Inverse binomial transform of A091002.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,6).

Cf. A015441.

Concatenation([0], List([1..30], n -> (3*2^n + 2*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019

[0] cat [(3*2^n + 2*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019

a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Join[{0, 0}, LinearRecurrence[{-1, 6}, {1, -1}, 30]] (* G. C. Greubel, Feb 01 2019 *)
CoefficientList[Series[x^2/((1-2x)(1+3x)),{x,0,30}],x] (* Harvey P. Dale, Apr 30 2022 *)

vector(30, n, n--; (3*2^n + 2*(-3)^n - 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019

[0] + [(3*2^n + 2*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

2^n = A091003(n) + 3*A091004(n) + 6*a(n).
a(n) = (3*2^n + 2*(-3)^n - 5*0^n)/30.
E.g.f.: (3*exp(2*x) + 2*exp(-3*x) - 5)/30. - G. C. Greubel, Feb 01 2019

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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A166035 a(n) = (3^n+6*(-4)^n)/7.

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A166036 a(n) = (4^n+8*(-5)^n)/9.

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A091003 Expansion of (1-3*x^2)/((1-2*x)*(1+3*x)).

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A091005 Expansion of x^2/((1-2*x)*(1+3*x)).

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GAP

Magma

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A091003 Expansion of (1-3x^2)/((1-2x)(1+3x)).

A091005 Expansion of x^2/((1-2x)(1+3*x)).