A165747 -id:A165747 - 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)

A192011 Let P(0,x) = -1, P(1,x) = 2x, and P(n,x) = xP(n-1,x) - P(n-2,x) for n > 1. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

Original entry on oeis.org

-1, 2, 0, 2, 0, 1, 2, 0, -1, 0, 2, 0, -3, 0, -1, 2, 0, -5, 0, 0, 0, 2, 0, -7, 0, 3, 0, 1, 2, 0, -9, 0, 8, 0, 1, 0, 2, 0, -11, 0, 15, 0, -2, 0, -1, 2, 0, -13, 0, 24, 0, -10, 0, -2, 0, 2, 0, -15, 0, 35, 0, -25, 0, 0, 0, 1, 2, 0, -17, 0, 48, 0, -49, 0, 10, 0, 3, 0, 2, 0, -19, 0, 63, 0, -84, 0, 35, 0, 3, 0, -1

Offset: 0

Paul Curtz, Jun 21 2011

			The first few rows are
  -1;
   2,   0;
   2,   0,   1;
   2,   0,  -1,   0;
   2,   0,  -3,   0,  -1;
   2,   0,  -5,   0,   0,   0;
   2,   0,  -7,   0,   3,   0,   1;
   2,   0,  -9,   0,   8,   0,   1,   0;
   2,   0, -11,   0,  15,   0,  -2,   0,  -1;
   2,   0, -13,   0,  24,   0, -10,   0,  -2,   0;
   2,   0, -15,   0,  35,   0, -25,   0,   0,   0,   1;

G. C. Greubel, Rows n = 0..30 of triangle, flattened

Left hand diagonals are: T(n,0) = [-1,2,2,2,2,2,...], T(n,2) = A165747(n), T(n,4) = A067998(n+1), T(n,6) = -A058373(n), T(n,8) = (-1)^(n+1) * A167387(n+2) (see also A052472(n)).

A192011 := proc(n,k)
        option remember;
        if k>n or k <0 or n<0 then
                0;
        elif n= 0 then
                -1;
        elif k=0 then
                2;
        else
                procname(n-1,k)-procname(n-2,k-2) ;
        end if;
end proc: # R. J. Mathar, Nov 03 2011

p[0, ] = -1; p[1, x] := 2x; p[n_, x_] := p[n, x] = x*p[n-1, x] - p[n-2, x]; row[n_] := CoefficientList[p[n, x], x]; Table[row[n] // Reverse, {n, 0, 9}] // Flatten (* Jean-François Alcover, Nov 26 2012 *)
T[n_,k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, -1, If[k==0, 2, T[n-1,k] - T[n-2, k-2]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 19 2019 *)

{T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, -1, if(k==0, 2, T(n-1,k) - T(n-2,k-2)))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, May 19 2019

def T(n,k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return -1
    elif (k==0): return 2
    else: return T(n-1,k) - T(n-2, k-2)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 19 2019

T(n, k) = T(n-1, k) - T(n-2, k-2), where T(0, 0) = -1, T(n, 0) = 2 and 0 <= k <= n, n >= 0. - G. C. Greubel, May 19 2019

A159964 a(n) = 2^n*(1-n).

Original entry on oeis.org

1, 0, -4, -16, -48, -128, -320, -768, -1792, -4096, -9216, -20480, -45056, -98304, -212992, -458752, -983040, -2097152, -4456448, -9437184, -19922944, -41943040, -88080384, -184549376, -385875968, -805306368, -1677721600, -3489660928

Offset: 0

Paul Barry, Apr 28 2009

Hankel transform of A124791. Binomial transform of -A060747.

{1} U A159964 is a composition of generating functions of A165747 and A000012, with H=G(F(x)) with F(x) for A000012 and G(x) for A165747. - Oboifeng Dira, Aug 29 2019

Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A058922, A060747, A124791.

LinearRecurrence[{4,-4},{1,0},30] (* Harvey P. Dale, May 02 2016 *)

G.f.: (1-4x)/(1-2x)^2.
a(n) = -A058922(n). - Jeffrey R. Goodwin, Nov 11 2011
E.g.f.: U(0) where U(k)=  1 - 2*x/(2 - 4/(2 - (k+1)/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 18 2012
a(n) = Sum_{k=0..n} (1-2k) * C(n,k). - Wesley Ivan Hurt, Sep 23 2017
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=2} 1/a(n) = -log(2)/2.
Sum_{n>=2} (-1)^n/a(n) = -log(3/2)/2. (End)

A144704 a(n) = 4^n(1-2n).

Original entry on oeis.org

1, -4, -48, -320, -1792, -9216, -45056, -212992, -983040, -4456448, -19922944, -88080384, -385875968, -1677721600, -7247757312, -31138512896, -133143986176, -566935683072, -2405181685760, -10170482556928

Offset: 0

Paul Barry, Sep 19 2008

With the n-th term of A000984 (C(2n,n)) as numerator, |a(n)| is the denominator of the probability that a random walk with steps of +-1 will return to the starting point for the first time after 2n steps. - Shel Kaphan, Jan 12 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Hankel transform of A100320.

Cf. A000302, A165747.

LinearRecurrence[{8,-16},{1,-4},30] (* Harvey P. Dale, Jun 12 2019 *)

[4^n*(1-2*n) for n in (0..30)] # G. C. Greubel, Jun 16 2022

G.f.: (1-12*x)/(1-4*x)^2.
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=0} 1/a(n) = 1 - arctanh(1/2)/2.
Sum_{n>=0} (-1)^(n+1)/a(n) = 1 + arctan(1/2)/2. (End)
E.g.f.: (1 - 8*x)*exp(4*x). - G. C. Greubel, Jun 16 2022
Sum_{n >= 1} x^(2*n-1)/a(n) = 1/4 * log((1 - x/2)/(1 + x/2)). Eldar's two summations above follow from this on setting x = 1 and x = i. - Peter Bala, Jul 08 2024

A098875 Decimal expansion of Sum_{n>0} n/exp(n).

Original entry on oeis.org

9, 2, 0, 6, 7, 3, 5, 9, 4, 2, 0, 7, 7, 9, 2, 3, 1, 8, 9, 4, 5, 4, 1, 3, 5, 2, 2, 7, 1, 6, 4, 9, 9, 6, 0, 2, 8, 8, 1, 6, 5, 5, 6, 2, 6, 6, 5, 0, 5, 5, 1, 1, 5, 2, 3, 5, 3, 9, 6, 0, 4, 0, 9, 7, 2, 2, 0, 4, 7, 1, 9, 7, 4, 6, 5, 0, 2, 4, 4, 5, 6, 8, 6, 7, 3, 6, 9, 9, 7, 3, 2, 8, 3, 4, 3, 4, 7, 9, 4, 7, 2, 5, 3, 9, 7

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004

The expression generating this constant is a first degree Eulerian polynomial, in the "variable" e, with coefficient {1}, generated from sum_{n>=0} n^m/e^n, with m=1. See A008292. It approximates m!. - Richard R. Forberg, Feb 15 2015

See A255169 for the second degree polynomial and value.

			0.9206735942077923189454135227164996028816556266505511523539604097220...

Cf. A001113, A008292, A010050, A027641, A027642, A073333, A165747, A255169.

g:=x->sum(n/exp(n),n=1..x); evalf[110](g(1500)); evalf[110](g(4000));

RealDigits[E/(E-1)^2, 10, 105][[1]] (* Jean-François Alcover, Jan 28 2014 *)

1+sumalt(n=1,bernreal(2*n)*(1-2*n)/(2*n)!) \\ Gleb Koloskov, Jul 12 2021

Equals exp(1)/(exp(1)-1)^2.
From Gleb Koloskov, Jul 12 2021: (Start)
Equals (1/2)/(cosh(1)-1).
Equals 1+Sum_{n>0} B(2*n)*(1-2*n)/(2*n)! = 1+Sum_{n>0} (A027641(2*n)/A027642(2*n))*A165747(n)/A010050(n).
Equals LambertW(x)*LambertW(-1,x), where x = (1/(1-e))*exp(1/(1-e)) = -A073333*exp(-A073333). (End)

A330885 Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, -1, 1, 4, 5, 0, -3, 1, 5, 7, 1, -5, -3, 1, 6, 9, 2, -7, -8, 1, 1, 7, 11, 3, -9, -13, -3, 7, 1, 8, 13, 4, -11, -18, -7, 10, 9, 1, 9, 15, 5, -13, -23, -11, 13, 21, 1, 1, 10, 17, 6, -15, -28, -15, 16, 33, 14, -15

Offset: 0

Bob Selcoe, May 05 2020

			Array starts:
1  1   1  -1   -3   -3    1   7   9   1  -15   -25
1  2   3   0   -5   -8   -3  10  21  14  -17   -52
1  3   5   1   -7  -13   -7  13  33  27  -19   -79
1  4   7   2   -9  -18  -11  16  45  40  -21  -106
1  5   9   3  -11  -23  -15  19  57  53  -23  -133
1  6  11   4  -13  -28  -19  22  69  66  -25  -160
1  7  13   5  -15  -33  -23  25  81  79  -27  -187

Cf. A078016, A180735.

Columns k: A000012 (k=0), A000027 (k=1), A005408 (k=2), A023443 (k=3), A165747 (k=4), -A016885 (k=5), -A004767 (k=6), A016777 (k=7), A017629 (k=8), A190991 (k=9).

T[n_, k_]:= T[n, k]= If[k<3, k*n+1, T[n, k-1] - T[n, k-2] - T[n, k-3]];
Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2020 *)

T(0,k) = A180735(k-1).

T(n,k) - T(n-1,k) = -A078016(k+1).

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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A192011 Let P(0,x) = -1, P(1,x) = 2*x, and P(n,x) = x*P(n-1,x) - P(n-2,x) for n > 1. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

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A159964 a(n) = 2^n*(1-n).

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A144704 a(n) = 4^n*(1-2*n).

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A098875 Decimal expansion of Sum_{n>0} n/exp(n).

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A330885 Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

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A192011 Let P(0,x) = -1, P(1,x) = 2x, and P(n,x) = xP(n-1,x) - P(n-2,x) for n > 1. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

A144704 a(n) = 4^n(1-2n).