A025564 -id:A025564 - 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

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878

Offset: 0

Clark Kimberling

nonn

a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006

Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
C. Dalfó, M. A. Fiol, and N. López, New results for the Mondrian art problem, arXiv:2007.09639 [math.CO], 2020.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

First differences of A026135. Row sums of triangle A026105.
Pairwise sums of A005727. Column k=2 in A115990.
Cf. A001791, A132816.
Cf. A136537.

List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # Muniru A Asiru, Mar 09 2019

seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002
Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007
a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007
D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

A025568 a(n) = T(n,n+2) where T is the array defined in A025564.

Original entry on oeis.org

1, 5, 19, 65, 211, 665, 2058, 6294, 19095, 57607, 173096, 518596, 1550367, 4627455, 13795176, 41088456, 122297643, 363828663, 1081966875, 3216725841, 9561635853, 28418162003, 84455354206, 250982289650, 745860104145, 2216567725281

Offset: 1

Clark Kimberling

nonn

First differences are pairwise sums of A025181.

Pairwise sums of A014532.

A025567 a(n) = T(n,n+1), where T is the array defined in A025564.

Original entry on oeis.org

1, 4, 13, 40, 120, 356, 1050, 3088, 9069, 26620, 78133, 229384, 673699, 1979628, 5820195, 17121312, 50394579, 148413996, 437324919, 1289330520, 3803175474, 11223840012, 33139076292, 97889042384, 289276841475, 855205791076, 2529279459099

Offset: 1

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5

Pairwise sums of A014531.

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+3];
Array[a, 27] (* Jean-François Alcover, Oct 30 2018 *)

x='x+O('x^66); Vec((x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3)) \\ Joerg Arndt, May 01 2013

G.f.: (x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3). - Mark van Hoeij, May 01 2013
Conjecture: (n+3)*a(n) +4*(-n-2)*a(n-1) +2*a(n-2) +8*(n-1)*a(n-3) +3*(n-3)*a(n-4)=0. - R. J. Mathar, Apr 03 2015
Conjecture: (n-1)*(n-2)*(n+3)*a(n) -2*n*(n-2)*(n+2)*a(n-1) -3*n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Apr 03 2015
a(n) ~ 2 * 3^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 02 2024

A025574 T(2n,n+1), where T is the array defined in A025564.

Original entry on oeis.org

1, 10, 61, 356, 2058, 11892, 68860, 399828, 2328066, 13591364, 79538750, 466489520, 2741310614, 16137748980, 95152639095, 561856988100, 3322001296650, 19664909435556, 116535288360106, 691284470658376, 4104450497833036, 24390520567203960, 145053186645586383

Offset: 1

Clark Kimberling

nonn

Coefficient of y^(n+1) in (y^2+2*y+1)*(y^2+y+1)^(2*n-2). - Robert Israel, Sep 07 2019

Robert Israel, Table of n, a(n) for n = 1..1278

Cf. A025564.

f:= n -> coeff((y^2+2*y+1)*(y^2+y+1)^(2*n-2),y,n+1);
map(f, [$1..30]); # Robert Israel, Sep 07 2019

More terms from Sean A. Irvine, Sep 07 2019

A025569 T(2n-1,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 4, 22, 120, 665, 3732, 21153, 120835, 694590, 4013088, 23284424, 135580865, 791862354, 4637009300, 27215510265, 160053426720, 942939231825, 5564024727096, 32878367902380, 194530139772844, 1152303896900338, 6832908809693304, 40556775927067299, 240938382783554100

Offset: 1

Clark Kimberling

nonn

Cf. A025564.

T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)));
a(n) = T(2*n, n+1); \\ Michel Marcus, Sep 08 2019

a(19) corrected and more terms from Sean A. Irvine, Sep 07 2019

A025570 a(n) = T(2n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 2, 8, 40, 211, 1148, 6369, 35816, 203424, 1164228, 6703372, 38785840, 225325940, 1313494396, 7679038865, 45006411960, 264357297375, 1555762562100, 9171399730944, 54148782940112, 320136500713070, 1895036382245032, 11230166960501463, 66619011999761208, 395563260817743219

Offset: 1

Clark Kimberling

nonn

More terms from Sean A. Irvine, Sep 07 2019

A025571 a(n) = T(3n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 3, 19, 140, 1090, 8749, 71604, 593997, 4976785, 42016975, 356879250, 3046096983, 26105109424, 224485080580, 1936029865870, 16738879226139, 145041018993018, 1259189426140919, 10950454909118725, 95374995078513150, 831822673255797630, 7263789225782699928

Offset: 1

Clark Kimberling

nonn

More terms from Sean A. Irvine, Sep 07 2019

A025572 a(n) = T(4n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 4, 34, 330, 3381, 35700, 384307, 4193020, 46203670, 513011312, 5730482707, 64324536822, 724966763955, 8198552413800, 92985865982415, 1057267681773972, 12047638469222921, 137547814582468048, 1573060924386848400, 18017644508154675426, 206653990802794950415

Offset: 1

Clark Kimberling

nonn

More terms from Sean A. Irvine, Sep 07 2019

A025573 a(n) = T(2n,n-1), where T is the array defined in A025564.

Original entry on oeis.org

1, 4, 19, 98, 526, 2892, 16159, 91338, 520794, 2989688, 17256275, 100046894, 582204290, 3398722420, 19894375665, 116724946350, 686260868346, 4042058735880, 23846068472626, 140882925528892, 833421929615236, 4936088399361876, 29266184138206509, 173689743532678182

Offset: 1

Clark Kimberling

nonn

More terms from Sean A. Irvine, Sep 07 2019

cp's OEIS Frontend

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

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A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

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A025568 a(n) = T(n,n+2) where T is the array defined in A025564.

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A025567 a(n) = T(n,n+1), where T is the array defined in A025564.

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A025574 T(2n,n+1), where T is the array defined in A025564.

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A025569 T(2n-1,n), where T is the array defined in A025564.

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A025570 a(n) = T(2n,n), where T is the array defined in A025564.

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A025571 a(n) = T(3n,n), where T is the array defined in A025564.

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A025572 a(n) = T(4n,n), where T is the array defined in A025564.

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A025573 a(n) = T(2n,n-1), where T is the array defined in A025564.

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