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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
 1, 0, 2, 6, 22, 80, 296, 1106, 4166, 15792, 60172, 230252, 884236, 3406104, 13154948, 50922986, 197519942, 767502944, 2987013068, 11641557716, 45429853652, 177490745984, 694175171648, 2717578296116, 10648297329692, 41757352712480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A Catalan transform of A078008 under the mapping g(x)->g(xc(x)). - Paul Barry, Nov 13 2004 a(n) = A108561(2*(n-1),n-1). - Reinhard Zumkeller, Jun 10 2005 Number of positive terms in expansion of (x_1 + x_2 + ... + x_{n-1} - x_n)^n. - Sergio Falcon, Feb 08 2007 Hankel transform is A088138(n+1). - Paul Barry, Feb 17 2009 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 Also central terms of triangles A108561 and A112465. - Reinhard Zumkeller, Jan 03 2014 REFERENCES L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016. Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, 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.] Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5. Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019. FORMULA If offset is 0, a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k). - Vladeta Jovovic, Feb 18 2003 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 a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-4, n-3). - Emeric Deutsch, Jan 28 2004 a(n) = (-1)^n*Sum_{k=0..n} binomial(-n,k) (offset 0). - Paul Barry, Feb 17 2009 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 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 From Peter Bala, Oct 01 2015: (Start) a(n) = [x^n] ((1 - x)^2/(1 - 2*x))^n. 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) EXAMPLE The array begins: 1 0 1 0 1.. 0 0 1 1 2.. 1 1 2 3 5.. 0 1 3 6 11.. MAPLE 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 MATHEMATICA 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 *) PROG (Haskell) a072547 n = a108561 (2 * (n - 1)) (n - 1) -- Reinhard Zumkeller, Jan 03 2014 (PARI) a(n) = (-1)^n*sum(k=0, n, binomial(-n, k)); vector(100, n, a(n-1)) \\ Altug Alkan, Oct 02 2015 (MAGMA) R:=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 (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 CROSSREFS Cf. A014300, A026641, A092785, A000957. Cf. A026641, A172025, A172061, A172062, A172063, A172064, A172065, A172066. - Richard Choulet, Jan 25 2010 Sequence in context: A203038 A206304 A201372 * A150229 A150230 A191755 Adjacent sequences:  A072544 A072545 A072546 * A072548 A072549 A072550 KEYWORD nonn AUTHOR Benoit Cloitre, Aug 05 2002 EXTENSIONS Corrected and extended by Vladeta Jovovic, Feb 17 2003 STATUS approved

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Last modified November 17 08:39 EST 2019. Contains 329217 sequences. (Running on oeis4.)