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A072547 Main diagonal of the array in which first column and row are filled alternatively with 1's or 0's and then T(i,j) = T(i-1,j) + T(i,j-1). 21

%I

%S 1,0,2,6,22,80,296,1106,4166,15792,60172,230252,884236,3406104,

%T 13154948,50922986,197519942,767502944,2987013068,11641557716,

%U 45429853652,177490745984,694175171648,2717578296116,10648297329692,41757352712480

%N Main diagonal of the array in which first column and row are filled alternatively with 1's or 0's and then T(i,j) = T(i-1,j) + T(i,j-1).

%C A Catalan transform of A078008 under the mapping g(x)->g(xc(x)). - _Paul Barry_, Nov 13 2004

%C a(n) = A108561(2*(n-1),n-1). - _Reinhard Zumkeller_, Jun 10 2005

%C Number of positive terms in expansion of (x_1 + x_2 + ... + x_{n-1} - x_n)^n. - _Sergio Falcon_, Feb 08 2007

%C Hankel transform is A088138(n+1). - _Paul Barry_, Feb 17 2009

%C Without the beginning "1", we obtain the first diagonal over the principal diagonal of the array notified by B. Cloitre in A026641 and used by R. Choulet in A172025, and from A172061 to A172066. - _Richard Choulet_, Jan 25 2010

%C Also central terms of triangles A108561 and A112465. - _Reinhard Zumkeller_, Jan 03 2014

%D L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

%H Reinhard Zumkeller, <a href="/A072547/b072547.txt">Table of n, a(n) for n = 1..1000</a>

%H David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, <a href="http://arxiv.org/abs/1605.06825">Pattern Avoiding Linear Extensions of Rectangular Posets</a>, arXiv:1605.06825 [math.CO], 2016.

%H Roland Bacher, <a href="http://arxiv.org/abs/1509.09054">Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle</a>, arXiv:1509.09054 [math.CO], 2015. [It is only a conjecture that this is the same sequence. It would be nice to have a proof.]

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.

%H Colin Defant, <a href="https://arxiv.org/abs/1905.02309">Proofs of Conjectures about Pattern-Avoiding Linear Extensions</a>, arXiv:1905.02309 [math.CO], 2019.

%H S. B. Ekhad, M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt"> Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)

%F If offset is 0, a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k). - _Vladeta Jovovic_, Feb 18 2003

%F G.f.: x*(1-x*C)/(1-2*x*C)/(1+x*C), where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers (A000108). - _Vladeta Jovovic_, Feb 18 2003

%F a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-4, n-3). - _Emeric Deutsch_, Jan 28 2004

%F a(n) = (-1)^n*Sum_{k=0..n} binomial(-n,k) (offset 0). - _Paul Barry_, Feb 17 2009

%F Other form of the G.f: f(z) = (2/(3*sqrt(1-4*z) -1 +4*z))*((1 -sqrt(1-4*z))/(2*z))^(-1). - _Richard Choulet_, Jan 25 2010

%F Conjecture: 2*(-n+1)*a(n) +(9*n-17)*a(n-1) +(-3*n+19)*a(n-2) +2*(-2*n+7) *a(n-3) = 0. - _R. J. Mathar_, Nov 30 2012

%F From _Peter Bala_, Oct 01 2015: (Start)

%F a(n) = [x^n] ((1 - x)^2/(1 - 2*x))^n.

%F Exp( Sum_{n >= 1} a(n+1)*x^n/n ) = 1 + x^2 + 2*x^3 + 6*x^4 + 18*x^5 + ... is the o.g.f for A000957. (End)

%e The array begins:

%e 1 0 1 0 1..

%e 0 0 1 1 2..

%e 1 1 2 3 5..

%e 0 1 3 6 11..

%p taylor( (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-1),z=0,42); for n from -1 to 40 do a(n):=sum('(-1)^(p)*binomial(2n-p+1,1+n-p)',p=0..n+1): od:seq(a(n),n=-1..40):od; # _Richard Choulet_, Jan 25 2010

%t CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x]) /(2*x))^(-1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)

%o (Haskell)

%o a072547 n = a108561 (2 * (n - 1)) (n - 1)

%o -- _Reinhard Zumkeller_, Jan 03 2014

%o (PARI) a(n) = (-1)^n*sum(k=0, n, binomial(-n, k));

%o vector(100, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015

%o (MAGMA) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( x*(1 + Sqrt(1-4*x))/(Sqrt(1-4*x)*(3-Sqrt(1-4*x))) )); // _G. C. Greubel_, Feb 17 2019

%o (Sage) a=(x*(1+sqrt(1-4*x))/(sqrt(1-4*x)*(3-sqrt(1-4*x)))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 17 2019

%Y Cf. A014300, A026641, A092785, A000957.

%Y Cf. A026641, A172025, A172061, A172062, A172063, A172064, A172065, A172066. - _Richard Choulet_, Jan 25 2010

%K nonn

%O 1,3

%A _Benoit Cloitre_, Aug 05 2002

%E Corrected and extended by _Vladeta Jovovic_, Feb 17 2003

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Last modified December 12 12:50 EST 2019. Contains 329958 sequences. (Running on oeis4.)