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).
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
Offset: 1
Keywords
Examples
The array begins: 1 0 1 0 1.. 0 0 1 1 2.. 1 1 2 3 5.. 0 1 3 6 11.. so sequence begins : 1, 0, 2, 6, ...
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, and 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.
- Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
- Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
Crossrefs
Programs
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Haskell
a072547 n = a108561 (2 * (n - 1)) (n - 1) -- Reinhard Zumkeller, Jan 03 2014
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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 -
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 *) 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 *) -
PARI
a(n) = (-1)^n*sum(k=0, n, binomial(-n, k)); vector(100, n, a(n-1)) \\ Altug Alkan, Oct 02 2015
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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
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) = A108561(2*(n-1),n-1). - Reinhard Zumkeller, Jun 10 2005
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
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
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)
a(n) = binomial(2*n-2, n)*hypergeom([1, 2-n], [n+1], 1/2) / 2 + (-2)^(1-n). - Peter Luschny, Dec 03 2021
a(n) = 2 * A014301(n-1) for n>=3. - Alois P. Heinz, Dec 27 2023
Extensions
Corrected and extended by Vladeta Jovovic, Feb 17 2003
Comments