A112467 - OEIS

A112467

Riordan array ((1-2x)/(1-x), x/(1-x)).

%I #66 Jul 07 2023 14:58:08

%S 1,-1,1,-1,0,1,-1,-1,1,1,-1,-2,0,2,1,-1,-3,-2,2,3,1,-1,-4,-5,0,5,4,1,

%T -1,-5,-9,-5,5,9,5,1,-1,-6,-14,-14,0,14,14,6,1,-1,-7,-20,-28,-14,14,

%U 28,20,7,1,-1,-8,-27,-48,-42,0,42,48,27,8,1,-1,-9,-35,-75,-90,-42,42,90,75,35,9,1,-1,-10,-44,-110,-165,-132,0,132,165,110

%N Riordan array ((1-2x)/(1-x), x/(1-x)).

%C Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.

%C (-1,1)-Pascal triangle. - _Philippe Deléham_, Aug 07 2006

%C Apart from initial term, same as A008482. - _Philippe Deléham_, Nov 07 2006

%C Each column equals the cumulative sum of the previous column. - _Mats Granvik_, Mar 15 2010

%C Reading along antidiagonals generates in essence rows of A192174. - _Paul Curtz_, Oct 02 2011

%C Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 01 2011

%H G. C. Greubel, <a href="/A112467/b112467.txt">Rows n = 0..100 of triangle, flattened</a>

%H Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, <a href="https://arxiv.org/abs/2307.02837">Restricting Dyck Paths and 312-avoiding Permutations</a>, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.

%H Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.

%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 Mathematics, 34 (2005) pp. 101-122.

%H D. Foata and G.-N. Han, <a href="http://dx.doi.org/10.1007/s11139-009-9194-9">The doubloon polynomial triangle</a>, Ram. J. 23 (2010), 107-126.

%H Jack Ramsay, <a href="/A349812/a349812.pdf">On Arithmetical Triangles</a>, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

%F Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).

%F Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - _Philippe Deléham_, Oct 03 2005

%F T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - _Roger L. Bagula_, Feb 16 2009; modified by _G. C. Greubel_, Dec 04 2019

%F T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - _Philippe Deléham_, Nov 01 2011

%F G.f.: (1-2x)/(1-(1+y)*x). - _Philippe Deléham_, Dec 15 2011

%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - _Philippe Deléham_, Dec 15 2011

%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 21 2014

%F Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - _G. C. Greubel_, Dec 04 2019

%e Triangle starts:

%e 1;

%e -1, 1;

%e -1, 0, 1;

%e -1, -1, 1, 1;

%e -1, -2, 0, 2, 1;

%e -1, -3, -2, 2, 3, 1;

%e -1, -4, -5, 0, 5, 4, 1;

%e -1, -5, -9, -5, 5, 9, 5, 1;

%e -1, -6, -14, -14, 0, 14, 14, 6, 1;

%e -1, -7, -20, -28, -14, 14, 28, 20, 7, 1;

%e -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1;

%e -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;

%e ...

%e From _Paul Barry_, Apr 08 2011: (Start)

%e Production matrix begins:

%e 1, 1,

%e -2, -1, 1,

%e 2, 0, -1, 1,

%e -2, 0, 0, -1, 1,

%e 2, 0, 0, 0, -1, 1,

%e -2, 0, 0, 0, 0, -1, 1,

%e 2, 0, 0, 0, 0, 0, -1, 1

%e ... (End)

%p seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # _G. C. Greubel_, Dec 04 2019

%t T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Roger L. Bagula_, Feb 16 2009; modified by _G. C. Greubel_, Dec 04 2019 *)

%o (PARI) T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ _G. C. Greubel_, Dec 04 2019

%o (Magma) [n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 04 2019

%o (Sage)

%o def T(n, k):

%o if (n==0): return 1

%o else: return (2*k-n)*binomial(n,k)/n

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 04 2019

%Y Same first 3 rows as in A054525.

%Y Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

%K easy,sign,tabl

%O 0,12

%A _Paul Barry_, Sep 06 2005