|
| |
|
|
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.
|
|
46
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,12
|
|
|
COMMENTS
|
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
|
|
|
LINKS
|
Paul D. Hanna, Table of n, a(n) for n = 0..1080
|
|
|
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.
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. - Philippe Deléham, Oct 07 2009
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. - Philippe Deléham, Oct 08 2009
|
|
|
EXAMPLE
|
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.
|
|
|
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 *)
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
Cf. A108561, A084247, A001045, A072547, A112556.
Cf. A112468 (reversed rows).
Cf. A140165. - Gary W. Adamson, Jan 30 2009
Sequence in context: A192062 A172371 A279006 * A108561 A174626 A264909
Adjacent sequences: A112552 A112553 A112554 * A112556 A112557 A112558
|
|
|
KEYWORD
|
sign,tabl
|
|
|
AUTHOR
|
Paul D. Hanna, Sep 21 2005
|
|
|
STATUS
|
approved
|
| |
|
|
