This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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<=k<=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<=k<=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 [From 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.