A105696 -id:A105696 - 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

A025178 First differences of the central trinomial coefficients A002426.

Original entry on oeis.org

0, 2, 4, 12, 32, 90, 252, 714, 2032, 5814, 16700, 48136, 139152, 403286, 1171380, 3409020, 9938304, 29017878, 84844044, 248382516, 727971360, 2135784798, 6272092596, 18435108258, 54228499920, 159636389850, 470256930052, 1386170197704

Offset: 1

Clark Kimberling

Previous name was: "a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0 = s(n), |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array defined in A025177."

Note that n-1 divides a(n) for n>=2. - T. D. Noe, Mar 16 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.

a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
seq(simplify(a(n)), n=1..28); # Peter Luschny, Oct 29 2015

Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* Harvey P. Dale, Aug 22 2011 *)
Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* Peter Luschny, Nov 03 2015 *)

a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ Altug Alkan, Oct 29 2015

def a():
    b, c, n = 0, 2, 2
    yield b
    while True:
        yield c
        b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
        n += 1
A025178 = a()
print([next(A025178) for  in (1..20)]) # _Peter Luschny, Nov 04 2015

a(n) = T(n,n) for n>=1, where T is the array defined in A025177.
a(n) = A002426(n+1) - A002426(n). - Benoit Cloitre, Nov 02 2002
a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - Benoit Cloitre, Nov 02 2002
a(n) = 2*(n-1)*A001006(n-2). - M. F. Hasler, Oct 24 2011
a(n) = 2*A005717(n-1). - R. J. Mathar, Jul 09 2012
E.g.f. Integral(Integral(2*exp(x)*((1-1/x)*BesselI(1,2*x) + 2*BesselI(0,2*x)))). - Sergei N. Gladkovskii, Aug 16 2012
G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - Sergei N. Gladkovskii, Aug 16 2012
D-finite with recurrence: a(n) = ((2+n)*a(n-2)+3*(3-n)*a(n-3)+3*(n-1)*a(n-1))/n, a(0)=1, a(1)=0, a(2)=2. - Sergei N. Gladkovskii, Aug 16 2012 [adapted to new offset by Peter Luschny, Nov 04 2015]
G.f.: (1-x)/x^2*G(0) - 1/x^2 , where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
From Peter Bala, Oct 28 2015: (Start)
a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.
n*(n-2)*a(n) = (2*n-3)*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2) with a(1) = 0, a(2) = 2. (End)
From Peter Luschny, Oct 29 2015: (Start)
a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).
a(n) = (n-1)*A007971(n) for n>=2.
A105696(n) = a(n-1) + a(n) for n>=2.
A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.
A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.
(End)

New name based on a comment by T. D. Noe, Mar 16 2005, offset set to 1 and a(1) = 0 prepended by Peter Luschny, Nov 04 2015

A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

David Applegate, Jun 29 2009

A010709 preceded by 1, 2.
Partial sums give A131098.
The INVERT transform gives A077996 without A077996(0). The Motzkin transform gives A105696 without A105696(0). Decimal expansion of 28/225=0.12444... . - R. J. Mathar, Jun 29 2009
Continued fraction expansion of 1 + sqrt(1/5). - Arkadiusz Wesolowski, Mar 30 2012
The number of solutions x (mod 2^(n+1)) of x^2 = 1 (mod 2^(n+1)), namely x = 1 (n=0), x = -1, 1 (n=1) and x = -1, 1, 2^n-1, 2^n+1 (n at least 2). - Christopher J. Smyth, May 15 2014
Also, the number of n-step self-avoiding walks on the L-lattice with no non-contiguous adjacencies (see A322419 for details of L-lattice). - Sean A. Irvine, Jul 29 2020

David Applegate, The movie version
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010709, A131098, A077996, A105696.

[ n le 1 select n+1 else 4: n in [0..104] ];

f[n_] := Fold[#2*Floor[#1/#2 + 1/2] &, n, Reverse@ Range[n - 1]]; Array[f, 55]

Vec((1+x+2*x^2)/(1-x) + O(x^100)) \\ Altug Alkan, Jan 19 2016

G.f.: (1+x+2*x^2)/(1-x).
E.g.f. A(x)=x*B(x) satisfies the differential equation B'(x)=1+x+x^2+B(x). - Vladimir Kruchinin, Jan 19 2011
E.g.f.: 4*exp(x) - 2*x - 3. - Elmo R. Oliveira, Aug 06 2024

A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

Original entry on oeis.org

2, 6, 26, 78, 242, 726, 2186, 6558, 19682, 59046, 177146, 531438, 1594322, 4782966, 14348906, 43046718, 129140162, 387420486, 1162261466, 3486784398, 10460353202, 31381059606, 94143178826, 282429536478, 847288609442, 2541865828326, 7625597484986, 22876792454958, 68630377364882, 205891132094646, 617673396283946, 1853020188851838, 5559060566555522, 16677181699666566

Offset: 1

Paul D. Hanna, Nov 26 2018

nonn

Triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n )/(1-y).

			G.f.: A(x) = 2*x + 6*x^2 + 26*x^3 + 78*x^4 + 242*x^5 + 726*x^6 + 2186*x^7 + 6558*x^8 + 19682*x^9 + 59046*x^10 + ...
L.g.f.: L(x)  =  log( (1-x)*(1-x^2)/(1-3*x) )  =  2*x + 6*x^2/2 + 26*x^3/3 + 78*x^4/4 + 242*x^5/5 + 726*x^6/6 + 2186*x^7/7 + 6558*x^8/8 + 19682*x^9/9 + 59046*x^10/10 + 177146*x^11/11 + ... + A321600(n,n-1)*x^n/n + ...
such that
exp(L(x)) = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 144*x^5 + 432*x^6 + 1296*x^7 + 3888*x^8 + 11664*x^9 + 34992*x^10 + 104976*x^11 + ... + A257970(n)*x^n + ...
exp(L(x)/2) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 44*x^5 + 122*x^6 + 342*x^7 + 966*x^8 + 2746*x^9 + 7846*x^10 + 22514*x^11 + 64836*x^12 + ... + A105696(n)*x^n + ...

Index entries for linear recurrences with constant coefficients, signature (3, 1, -3).

Cf. A321600, A257970, A105696.

{a(n) = n*polcoeff( log((1 - x)*(1 - x^2)/(1 - 3*x +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: log( (1 - x)*(1 - x^2)/(1 - 3*x) ).

G.f.: 2*x*(1 + 3*x^2)/((1 - x^2)*(1 - 3*x)).

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

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A025178 First differences of the central trinomial coefficients A002426.

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A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

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A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

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