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

1,3

Comments

A Catalan transform of A078008 under the mapping g(x)->g(xc(x)). - Paul Barry, Nov 13 2004

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

With offset 0 and for p prime, the p-th term is divisible by p. - F. Chapoton, Dec 03 2021

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)

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

Example

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, ...

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 *)

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<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
(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, A014301, 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 A360266

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