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%I #64 Aug 28 2022 08:40:31
%S 1,1,1,1,0,1,1,1,-1,1,1,0,2,-2,1,1,1,-2,4,-3,1,1,0,3,-6,7,-4,1,1,1,-3,
%T 9,-13,11,-5,1,1,0,4,-12,22,-24,16,-6,1,1,1,-4,16,-34,46,-40,22,-7,1,
%U 1,0,5,-20,50,-80,86,-62,29,-8,1,1,1,-5,25,-70,130,-166,148,-91,37,-9,1,1,0,6,-30,95,-200,296,-314,239,-128,46,-10,1
%N Riordan array (1/(1-x), x/(1+x)).
%C Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).
%C Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). 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. - _Paul D. Hanna_, Jan 20 2006
%C Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - _Philippe Deléham_, Feb 22 2014
%H Reinhard Zumkeller, <a href="/A112468/b112468.txt">Rows n = 0..125 of triangle, flattened</a>
%H H. Belbachir and F. Bencherif, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Belbachir/belbachir13.html">On some properties of bivariate Fibonacci and Lucas Polynomials</a>, JIS 11 (2008) 08.2.6.
%H Hacene Belbachir and Athmane Benmezai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Belbachir/bel22.html">Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
%H Emeric Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
%H Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H H. Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Prodinger/prodinger27.html">On the expansion of Fibonacci and Lucas Polynomials</a>, JIS 12 (2009) 09.1.6.
%F Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - _Mats Granvik_, Mar 15 2010
%F Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).
%F G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - _Paul D. Hanna_, Jan 20 2006
%F T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - _Peter Luschny_, Jul 25 2014
%e Triangle starts
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, 1, -1, 1;
%e 1, 0, 2, -2, 1;
%e 1, 1, -2, 4, -3, 1;
%e 1, 0, 3, -6, 7, -4, 1;
%e Matrix log begins:
%e 0;
%e 1, 0;
%e 1, 0, 0;
%e 1, 1, -1, 0;
%e 1, 1, 1, -2, 0;
%e 1, 1, 1, 1, -3, 0; ...
%e Production matrix begins
%e 1, 1,
%e 0, -1, 1,
%e 0, 0, -1, 1,
%e 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, -1, 1.
%e - _Paul Barry_, Apr 08 2011
%p T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)*hypergeom( [1,n+1],[k+2],m)/(k+1)!; A112468 := (n,k) -> T(n,n-k,-1);
%p seq(print(seq(simplify(A112468(n,k)),k=0..n)),n=0..10); # _Peter Luschny_, Jul 25 2014
%t T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Mar 06 2013 *)
%o (PARI) {T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x),n,X),k,Y)} \\ _Paul D. Hanna_, Jan 20 2006
%o (Haskell)
%o a112468 n k = a112468_tabl !! n !! k
%o a112468_row n = a112468_tabl !! n
%o a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
%o -- _Reinhard Zumkeller_, Jan 03 2014
%o (PARI) T(n,k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ _G. C. Greubel_, Nov 13 2019
%o (Magma)
%o function T(n,k)
%o if k eq 0 or k eq n then return 1;
%o else return T(n-1,k-1) - T(n-1,k);
%o end if;
%o return T;
%o end function;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 13 2019
%o (Sage)@CachedFunction
%o def T(n, k):
%o if (k<0 or n<0): return 0
%o elif (k==0 or k==n): return 1
%o else: return T(n-1, k-1) - T(n-1, k)
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 13 2019
%o (GAP)
%o T:= function(n,k)
%o if k=0 or k=n then return 1;
%o else return T(n-1,k-1) - T(n-1,k);
%o fi;
%o end;
%o Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 13 2019
%Y Cf. A174294, A174295, A174296, A174297. - _Mats Granvik_, Mar 15 2010
%Y Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.
%Y See A279006 for another version.
%K easy,sign,tabl
%O 0,13
%A _Paul Barry_, Sep 06 2005
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