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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).

%I A072547 #80 Dec 27 2023 09:02:15
%S A072547 1,0,2,6,22,80,296,1106,4166,15792,60172,230252,884236,3406104,
%T A072547 13154948,50922986,197519942,767502944,2987013068,11641557716,
%U A072547 45429853652,177490745984,694175171648,2717578296116,10648297329692,41757352712480
%N 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).
%C A072547 A Catalan transform of A078008 under the mapping g(x)->g(xc(x)). - _Paul Barry_, Nov 13 2004
%C A072547 Number of positive terms in expansion of (x_1 + x_2 + ... + x_{n-1} - x_n)^n. - _Sergio Falcon_, Feb 08 2007
%C A072547 Hankel transform is A088138(n+1). - _Paul Barry_, Feb 17 2009
%C A072547 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 A072547 Also central terms of triangles A108561 and A112465. - _Reinhard Zumkeller_, Jan 03 2014
%C A072547 With offset 0 and for p prime, the p-th term is divisible by p. - _F. Chapoton_, Dec 03 2021
%D A072547 L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
%H A072547 Reinhard Zumkeller, <a href="/A072547/b072547.txt">Table of n, a(n) for n = 1..1000</a>
%H A072547 David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, and 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 A072547 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 A072547 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 A072547 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A072547 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 A072547 S. B. Ekhad and 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 A072547 If offset is 0, a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k). - _Vladeta Jovovic_, Feb 18 2003
%F A072547 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 A072547 a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-4, n-3). - _Emeric Deutsch_, Jan 28 2004
%F A072547 a(n) = A108561(2*(n-1),n-1). - _Reinhard Zumkeller_, Jun 10 2005
%F A072547 a(n) = (-1)^n*Sum_{k=0..n} binomial(-n,k) (offset 0). - _Paul Barry_, Feb 17 2009
%F A072547 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 A072547 D-finite with recurrence 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 A072547 From _Peter Bala_, Oct 01 2015: (Start)
%F A072547 a(n) = [x^n] ((1 - x)^2/(1 - 2*x))^n.
%F A072547 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)
%F A072547 a(n) = binomial(2*n-2, n)*hypergeom([1, 2-n], [n+1], 1/2) / 2 + (-2)^(1-n). - _Peter Luschny_, Dec 03 2021
%F A072547 a(n) = 2 * A014301(n-1) for n>=3. - _Alois P. Heinz_, Dec 27 2023
%e A072547 The array begins:
%e A072547   1 0 1 0 1..
%e A072547   0 0 1 1 2..
%e A072547   1 1 2 3 5..
%e A072547   0 1 3 6 11..
%e A072547 so sequence begins : 1, 0, 2, 6, ...
%p A072547 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 A072547 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 *)
%t A072547 a[n_] := Binomial[2 n - 2, n] Hypergeometric2F1[1, 2 - n, n + 1, 1/2] / 2 + (-2)^(1 - n); Table[a[n], {n, 1, 26}] (* _Peter Luschny_, Dec 03 2021 *)
%o A072547 (Haskell)
%o A072547 a072547 n = a108561 (2 * (n - 1)) (n - 1)
%o A072547 -- _Reinhard Zumkeller_, Jan 03 2014
%o A072547 (PARI) a(n) = (-1)^n*sum(k=0, n, binomial(-n, k));
%o A072547 vector(100, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015
%o A072547 (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 A072547 (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 A072547 Cf. A014300, A014301, A026641, A092785, A000957.
%Y A072547 Cf. A026641, A172025, A172061, A172062, A172063, A172064, A172065, A172066. - _Richard Choulet_, Jan 25 2010
%K A072547 nonn
%O A072547 1,3
%A A072547 _Benoit Cloitre_, Aug 05 2002
%E A072547 Corrected and extended by _Vladeta Jovovic_, Feb 17 2003