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A109466 Riordan array (1, x(1-x)). 52
1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Inverse is Riordan array (1, xc(x)) (A106566).

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009

LINKS

Table of n, a(n) for n=0..94.

P. Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.

T. Copeland, Addendum to Elliptic Lie Triad

FORMULA

Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).

T(n, k)*2^(n-k) = A110509(n, k); T(n, k)*3^(n-k) = A110517(n, k).

Sum_{k, 0<=k<=n}T(n,k)*A000108(k)=1 . - Philippe Deléham, Jun 11 2007

Sum_{k, 0<=k<=n}T(n,k)*A144706(k)=A082505(n+1). - Philippe Deléham, Oct 30 2008

Sum_{k, 0<=k<=n}T(n,k)*A002450(k)=A100335(n). - Philippe Deléham, Oct 30 2008

Sum_{k, 0<=k<=n}T(n,k)*A001906(k)=A100334(n). - Philippe Deléham, Oct 30 2008

Sum_{k, 0<=k<=n}T(n,k)*A015565(k)=A099322(n). - Philippe Deléham, Oct 30 2008

Sum_{k, 0<=k<=n}T(n,k)*A003462(k)=A106233(n). - Philippe Deléham, Oct 30 2008

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008

G.f.: 1/(1-y*x+y*x^2). - Philippe Deléham, Dec 15 2011

T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(n,0) = 0^n. - Philippe Deléham, Feb 15 2012

Sum_[k, 0<=k<=n}T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013

Sum_{k, 0<=k<=n}T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 26 2013

Sum_{k, 0<=k<=n}T(n,k)*T(n+1,k) = -A110320(n). - Philippe Deléham, Feb 26 2013

For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016

EXAMPLE

Rows begin:

1;

0, 1;

0, -1, 1;

0, 0, -2, 1;

0, 0, 1, -3, 1;

0, 0, 0, 3, -4, 1;

0, 0, 0, -1, 6, -5, 1;

0, 0, 0, 0, -4, 10, -6, 1;

0, 0, 0, 0, 1, -10, 15, -7, 1;

0, 0, 0, 0, 0, 5, -20, 21, -8, 1;

0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1;

From Paul Barry, Sep 28 2009: (Start)

Production array is

0, 1,

0, -1, 1,

0, -1, -1, 1,

0, -2, -1, -1, 1,

0, -5, -2, -1, -1, 1,

0, -14, -5, -2, -1, -1, 1,

0, -42, -14, -5, -2, -1, -1, 1,

0, -132, -42, -14, -5, -2, -1, -1, 1,

0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)

PROG

(MAGMA) /* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016

CROSSREFS

Cf. A026729 (unsigned version), A000108, A30528, A124644.

Sequence in context: A108063 A164846 A026729 * A259095 A076833 A071676

Adjacent sequences:  A109463 A109464 A109465 * A109467 A109468 A109469

KEYWORD

easy,sign,tabl

AUTHOR

Philippe Deléham, Aug 28 2005

STATUS

approved

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Last modified December 10 15:32 EST 2016. Contains 279003 sequences.