A120580 -id:A120580 - cp's OEIS frontend

A025565 a(n) = T(n,n-1), where T is array defined in A025564.

1, 2, 4, 10, 26, 70, 192, 534, 1500, 4246, 12092, 34606, 99442, 286730, 829168, 2403834, 6984234, 20331558, 59287740, 173149662, 506376222, 1482730098, 4346486256, 12754363650, 37461564504, 110125172682, 323990062452, 953883382354

Offset: 1

Clark Kimberling

nonn

a(n+1) is the number of UDU-free paths of n upsteps (U) and n downsteps (D), n>=0. - David Callan, Aug 19 2004
Hankel transform is A120580. - Paul Barry, Mar 26 2010
If interpreted with offset 0, the inverse binomial transform of A006134 - Gary W. Adamson, Nov 10 2007
Also the number of different integer sets { k_1, k_2, ..., k_(i+1) } with Sum_{j=1..i+1} k_j = i and k_j >= 0, see the "central binomial coefficients" (A000984), without all sets in which any two successive k_j and k_(j+1) are zero. See the partition problem eq. 3.12 on p. 19 in my dissertation below. - Eva Kalinowski, Oct 18 2012

			G.f. = x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 70*x^6 + 192*x^7 + 534*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
D. Baccherini, D. Merlini, and R. Sprugnoli, Binary words excluding a pattern and proper Riordan arrays, Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See page 1034. - _N. J. A. Sloane_, Mar 25 2014
J. L. Jacobsen and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760; arXiv:cond-mat/0407444, 2004-2006. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.

Cf. A005773, A025564.
First column of A097692.
Partial sums of A105696.

a025565 n = a025565_list !! (n-1)
a025565_list = 1 : f a001006_list [1] where
   f (x:xs) ys = y : f xs (y : ys) where
     y = x + sum (zipWith (*) a001006_list ys)
-- Reinhard Zumkeller, Mar 30 2012

seq( add(binomial(i-2, k)*(binomial(i-k, k+1)), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# Alternatively:
a := n -> `if`(n=1,1,2*(-1)^n*hypergeom([3/2, 2-n], [2], 4)):
seq(simplify(a(n)),n=1..28); # Peter Luschny, Jan 30 2017

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
a[n_] := T[n-1, n-1];
Array[a, 30] (* Jean-François Alcover, Jul 30 2018 *)

def A():
    a, b, n  = 1, 1, 1
    yield a
    while True:
        yield a + b
        n += 1
        a, b = b, ((3*(n-1))*a+(2*n-1)*b)//n
A025565 = A()
print([next(A025565) for  in range(28)]) # _Peter Luschny, Jan 30 2017

G.f.: x*sqrt((1+x)/(1-3*x)).
a(n) = 2*A005773(n-1) for n > 1.
a(n) = |A085455(n-1)| = A025577(n) - A025577(n-1) = A002426(n) + A002426(n-1).
Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*a(j)*a(i-j) = 3^n. - Mario Catalani (mario.catalani(AT)unito.it), Jul 02 2003
a(1) = 1, a(n) = M(n-1) + Sum_{k=1..n-1} M(k-1)*a(n-k) with M=A001006, the Motzkin Numbers. - Reinhard Zumkeller, Mar 30 2012
D-finite with recurrence: (-n+1)*a(n) +2*(n-1)*a(n-1) +3*(n-3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
G.f.: G(0), where G(k) = 1 + 4*x*(4*k+1)/( (1+x)*(4*k+2) - x*(1+x)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1+x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = n*hypergeom([2-n, 1/2-n/2, 1-n/2], [2, -n], 4). - Peter Luschny, Jul 12 2016
a(n) = (-1)^n*2*hypergeom([3/2, 2-n], [2], 4) for n > 1. - Peter Luschny, Jan 30 2017

Incorrect statement related to A000984 (see A002426) and duplicate of the g.f. removed by R. J. Mathar, Oct 16 2009

Edited by R. J. Mathar, Aug 09 2010

A098479 Expansion of 1/sqrt((1-x)^2 - 4*x^3).

Original entry on oeis.org

1, 1, 1, 3, 7, 13, 27, 61, 133, 287, 633, 1407, 3121, 6943, 15517, 34755, 77959, 175213, 394499, 889461, 2007963, 4538485, 10269247, 23258881, 52726599, 119627977, 271624315, 617180533, 1403272799, 3192557561, 7267485523, 16552454205, 37718893317, 85992506271

Offset: 0

Paul Barry, Sep 10 2004

1/sqrt((1-x)^2-4*r*x^3) expands to Sum_{k=0..floor(n/2)} binomial(n-k,k)*binomial(n-2*k,k)*r^k.
Hankel transform is A120580. - Paul Barry, Sep 19 2008
From Joerg Arndt, Jul 01 2011:  (Start)
Apparently the number of lattice paths from (0,0) to (n,n) using steps (3,0), (0,3), (1,1).
It appears that 1/sqrt((1-x)^2-4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1).
Apparently the number of lattice paths from (0,0) to (n,n) using steps (1,2), (2,1), (1,1).  (End)
Diagonal of rational functions 1/(1 - (x*y + x*y^2 + x^2*y)), 1/(1 - (x*y + x^3 + y^3)). - Gheorghe Coserea, Aug 31 2018
Diagonal of the rational function 1 / ((1-x)*(1-y) - x^2*y^3). - Seiichi Manyama, Apr 29 2025

			From _Joerg Arndt_, Jul 01 2011: (Start)
The triangle of lattice paths from (0,0) to (n,k) using steps (1,2), (2,1), (1,1) begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 2, 3;
  0, 0, 1, 3, 7;
  0, 0, 0, 3, 7, 13;
  0, 0, 0, 1, 6, 17, 27;
  0, 0, 0, 0, 4, 14, 36, 61;
The triangle of lattice paths from (0,0) to (n,k) using steps (3,0), (0,3), (1,1) begins
  1;
  0, 1;
  0, 0, 1;
  1, 0, 0, 3;
  0, 2, 0, 0,  7;
  0, 0, 3, 0,  0, 13;
  1, 0, 0, 7,  0,  0, 27;
  0, 3, 0, 0, 17,  0,  0, 61;
The diagonals of both appear to be this sequence.  (End)

Michael De Vlieger, Table of n, a(n) for n = 0..2749
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
J. Cigler, Some nice Hankel determinants, arXiv preprint arXiv:1109.1449 [math.CO], 2011.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.

Cf. A098480, A098481.

a[n_] := Sum[ Binomial[n-k, k]*Binomial[n-2k, k], {k, 0, n/2}]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jan 07 2013, from 1st formula *)
CoefficientList[Series[1/Sqrt[(1-x)^2-4x^3],{x,0,40}],x] (* Harvey P. Dale, Aug 13 2024 *)

/* as lattice paths, assuming the first comment is true */
/* same as in A092566 but use either of the following */
steps=[[3,0], [0,3], [1,1]];
steps=[[1,1], [1,2], [2,1]];
/* Joerg Arndt, Jul 01 2011 */

from sympy import binomial
def a(n): return sum(binomial(n - k, k) * binomial(n - 2*k, k) for k in range(n//2 + 1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Apr 18 2017

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n-2*k, k).
D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + (n-1)*a(n-2) + 2*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
G.f.: 1/(1 - x - 2*x^3/(1 - x - x^3/(1 - x - x^3/(1 - x - x^3/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021
a(n) ~ 1 / (sqrt((1-r)*(3-r)) * sqrt(Pi*n) * r^n), where r = 0.432040800333095... is the real root of the equation -1 + 2*r - r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Jun 05 2022

A120582 Hankel transform of Sum_{k=0..floor(n/2)} binomial(2*k, k).

Original entry on oeis.org

1, 2, 0, 0, -16, -32, -64, -128, 0, 0, 1024, 2048, 4096, 8192, 0, 0, -65536, -131072, -262144, -524288, 0, 0, 4194304, 8388608, 16777216, 33554432, 0, 0, -268435456, -536870912, -1073741824, -2147483648, 0, 0, 17179869184, 34359738368, 68719476736, 137438953472, 0, 0, -1099511627776

Offset: 0

Paul Barry, Jun 15 2006

Hankel transform of A100066(n+1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-16).

Cf. A100066, A120580.

I:=[1,2,0,0]; [n le 4 select I[n] else 4*Self(n-2) -16*Self(n-4): n in [1..51]]; // G. C. Greubel, Jun 08 2022

LinearRecurrence[{0,4,0,-16}, {1,2,0,0}, 51] (* G. C. Greubel, Jun 08 2022 *)

def C(n): return floor(chebyshev_U(n, 1/2))
def A120582(n): return 2^n*( ((n+1)%2)*(C(n/2) - C((n-2)/2)) + (n%2)*C((n+1)/2) )
[A120582(n) for n in (0..50)] # G. C. Greubel, Jun 08 2022

a(n) = 2^n*(-sqrt(3)*cos(5*Pi*n/6 + Pi/3)/6 + (sqrt(3)/3 - 1/2)*sin(5*Pi*n/6 + Pi/3) + (sqrt(3)/3 + 1/2)*cos(Pi*n/6 + Pi/6) + sqrt(3)*sin(Pi*n/6 + Pi/6)/6).
G.f.: (1-2*x)*(1+2*x)^2/(1 - 4*x^2 + 16*x^4). - Colin Barker, Jun 27 2013
a(n) = 2^(n-1)*( (1+(-1)^n)*(ChebyshevU(n/2, 1/2) - ChebyshevU((n-2)/2, 1/2)) + (1 -(-1)^n)*ChebyshevU((n+1)/2, 1/2)). - G. C. Greubel, Jun 08 2022

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A025565 a(n) = T(n,n-1), where T is array defined in A025564.

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A098479 Expansion of 1/sqrt((1-x)^2 - 4*x^3).

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A120582 Hankel transform of Sum_{k=0..floor(n/2)} binomial(2*k, k).

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A120581 Hankel transform of sum{k=0..n, C(2k,k)*2^k}.

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