A127846 -id:A127846 - cp's OEIS frontend

A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785, 321327658068506121, 2703925940081270205

Offset: 0

Wenjin Woan, Jan 20 2001

If y = x*A(x) then 4*y^2 - (1 + 3*x)*y + x = 0 and x = y*(1 - 4*y) / (1 - 3*y). - Michael Somos, Sep 28 2003
a(n) = A059450(n, n). - Michael Somos, Mar 06 2004
The Hankel transform of this sequence is 4^binomial(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 3 colors. Example: a(2)=5 because we have UDUD, UUDD, UBD, UGD, and URD, where U=(1,1), D=(1,-1), while B, G, and R are, respectively, blue, green, and red (2,0)-steps. - Emeric Deutsch, May 02 2011
Shifts left when INVERT transform applied four times. - Benedict W. J. Irwin, Feb 02 2016

			a(3) = 29 since the top row of Q^2 = (5, 8, 16, 0, 0, 0, ...), and 5 + 8 + 16 = 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.8.
Zhi Chen and Hao Pan, Identities involving weighted Catalan, Schroder and Motzkin paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=1, b=4.
Curtis Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28 (the sequence d_n).
Curtis Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Joseph P. S. Kung and Anna de Mier, Catalan lattice paths with rook, bishop and spider steps, J. Comb. Theor., Series A (2013) Vol. 120, Issue 2, 379-389.
Gregory J. Morrow, Laws relating runs and steps in gambler’s ruin, Stochastic Proc. Appl. (2024) Vol. 125, Issue 5, 2010-2025.
Gregory Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See pp. 1, 4, 15, 22. DOI
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.
Wen-jin Woan, Diagonal lattice paths, Congr. Numer. 151 (2001) 173-178.

Cf. A000108, A001003, A007564, A127846.
Row sums of A086873.
Column k=2 of A227578. - Alois P. Heinz, Jul 17 2013

gf := (1+3*x-sqrt(9*x^2-10*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d,`,coeff(s, x, i)) od:
A059231_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+4*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A059231_list(20); # Peter Luschny, May 19 2011

Join[{1},Table[-I 3^n/2LegendreP[n,-1,5/3],{n,40}]] (* Harvey P. Dale, Jun 09 2011 *)
Table[Hypergeometric2F1[-n, 1 - n, 2, 4], {n, 0, 22}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x - sqrt(1 - 10*x + 9*x^2 + x^2 * O(x^n))) / (8*x), n))}; /* Michael Somos, Sep 28 2003 */

{a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Sep 28 2003 */

# Algorithm of L. Seidel (1877)
def A059231_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = False; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += 2*D[k+1]
        else :
            for k in range(h, 0, -1) : D[k] += 2*D[k-1]
            h += 1
        b = not b
        if b : R.append(D[1])
    return R
A059231_list(23)  # Peter Luschny, Oct 19 2012

a(n) = Sum_{k=0..n} 4^k*N(n, k) where N(n, k) = (1/n)*binomial(n, k)*binomial(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 10 2003
a(n) = 3^n/2*LegendreP(n, -1, 5/3). - Vladeta Jovovic, Sep 17 2003
G.f.: (1 + 3*x - sqrt(1 - 10*x + 9*x^2)) / (8*x) = 2 / (1 + 3*x + sqrt(1 - 10*x + 9*x^2)). - Michael Somos, Sep 28 2003
a(n) = Sum_{k=0..n} A088617(n, k)*4^k*(-3)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1: a(1)=1, a(n) = -3*a(n-1) + 4*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence a(n) = (5(2n-1)a(n-1) - 9(n-2)a(n-2))/(n+1) for n>=2; a(0)=a(1)=1. - Emeric Deutsch, Mar 20 2004
Moment representation: a(n)=(1/(8*Pi))*Int(x^n*sqrt(-x^2+10x-9)/x,x,1,9)+(3/4)*0^n. - Paul Barry, Sep 30 2009
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  4, 4, 4
  1, 1, 1, 1
  4, 4, 4, 4, 4
  1, 1, 1, 1, 1, 1
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
  1, 4, 0, 0, 0, ...
  1, 1, 4, 0, 0, ...
  1, 1, 1, 4, 0, ...
  1, 1, 1, 1, 4, ...
  ... - Gary W. Adamson, Aug 23 2011
G.f.: (1+3*x-sqrt(9*x^2-10*x+1))/(8*x)=(1+3*x -G(0))/(4*x) ; G(k)= 1+x*3-x*4/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ sqrt(2)*3^(2*n+1)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012
a(n) = A127846(n) for n>0. - Philippe Deléham, Apr 03 2013
0 = a(n)*(+81*a(n+1) - 225*a(n+2) + 36*a(n+3)) + a(n+1)*(+45*a(n+1) + 82*a(n+2) - 25*a(n+3)) + a(n+2)*(+5*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Aug 25 2014
G.f.: 1/(1 - x/(1 - 4*x/(1 - x/(1 - 4*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017

A182122 Expansion of exp( arcsinh( 2*x ) ).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280

Offset: 0

Michael Somos, Apr 13 2012

sign

			G.f. = 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A104624.

Cf. A000984, A048990, A078531, A224884, A245112, A245113.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(2*x)))); // G. C. Greubel, Aug 12 2018

s := proc(n) option remember; `if`(n<2, n+1, -4*(n-2)*s(n-2)/(n+1)) end: A127846 := n -> `if`(n<2,n+1,s(n-1)); seq(A127846(n), n=0..47); # Peter Luschny, Sep 23 2014

CoefficientList[Series[Exp[ArcSinh[2x]],{x,0,50}],x] (* Harvey P. Dale, Aug 18 2012 *)
Table[2 HypergeometricPFQ[{-n+1,2-n},{2},-1],{n,0,46}] (* Peter Luschny, Sep 23 2014 *)

{a(n) = if( n<2, (n>=0) + (n>0), n = n-2; if( n%2, 0, (-1)^(n/2) * 4 * binomial( n, n/2) / (n + 2)))};

{a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) + 2*x, n ) )};

{a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = sqrt( 1 + 4*x * A)); polcoeff( A, n))};

def A182122(n):
    if n < 2: return n+1
    if n % 2 == 1: return 0
    return (-1)^(n/2-1)*binomial(n,n/2)/(n-1)
[A182122(n) for n in range(47)] # Peter Luschny, Sep 23 2014

G.f.: 2*x + sqrt( 1 + 4*x^2 ) = 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x / ... )))).
The g.f. A(x) satisfies: A(x) = sqrt(1 + 4*x * A(x)).
a(n) = (-1)^n * A104624(n). Convolution inverse of A104624.
Conjecture : n*(n+1)*a(n) + (n+2)*(n-1)*a(n-1) +4*(n+1)*(n-3)*a(n-2) +4*(n+2)*(n-4)*a(n-3) = 0.- R. J. Mathar, Jul 24 2012
a(n) = 2*hypergeom([-n+1,2-n],[2],-1). - Peter Luschny, Sep 23 2014
0 = a(n)*(+16*a(n+2) + 10*a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+4)) if n>=0. - Michael Somos, Jan 10 2017
a(n+4) = 2 * a(n+2) * (a(n+2) - 8*a(n)) / (a(n+2) + 10*a(n)) if n>=0 is even. - Michael Somos, Jan 10 2017
G.f. A(x) satisfies A(x) = 1/A(-x). - Seiichi Manyama, Jun 20 2025

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7x+9x^2) ).

Original entry on oeis.org

1, 7, 58, 532, 5209, 53347, 564499, 6123481, 67732483, 761052565, 8662502212, 99671232514, 1157409133831, 13546774268125, 159649564550746, 1892849564159596, 22562032457415067, 270209749616920813, 3249905798884688038, 39237866746912398292, 475388228365424562019

Offset: 0

Seiichi Manyama, Aug 20 2025

nonn

Column k=3 of A386408.
Cf. A002212, A127846, A386389.
Cf. A000108, A337167, A385716.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+7*x+9*x^2))/x)

G.f.: 2/(1 - 7*x + sqrt((1-x) * (1-13*x))).
a(n) = (A337167(n+1) - A337167(n))/3.
(n+2)*a(n) = 7*(2*n+1)*a(n-1) - 13*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * Catalan(k+1).

A386389 Expansion of (1/x) * Series_Reversion( x/(1+9x+16x^2) ).

Original entry on oeis.org

1, 9, 97, 1161, 14849, 198729, 2748641, 38977353, 563644673, 8280210825, 123226850913, 1853870946057, 28148395838721, 430791367720905, 6638484468424929, 102918165951351753, 1604104541561284097, 25121009971212463881, 395085505395126968417, 6237523016309454855561

Offset: 0

Seiichi Manyama, Aug 20 2025

nonn

Column k=4 of A386408.
Cf. A002212, A127846, A386362.
Cf. A000108, A269796, A386387.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+9*x+16*x^2))/x)

G.f.: 2/(1 - 9*x + sqrt((1-x) * (1-17*x))).
a(n) = (A386387(n+1) - A386387(n))/4.
(n+2)*a(n) = 9*(2*n+1)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 16^k * 9^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k+1).

A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 29, 36, 1, 1, 9, 58, 185, 137, 1, 1, 11, 97, 532, 1257, 543, 1, 1, 13, 146, 1161, 5209, 8925, 2219, 1, 1, 15, 205, 2156, 14849, 53347, 65445, 9285, 1, 1, 17, 274, 3601, 34041, 198729, 564499, 491825, 39587, 1

Offset: 0

Seiichi Manyama, Aug 20 2025

			Square array begins:
  1,    1,     1,      1,       1,       1,        1, ...
  1,    3,     5,      7,       9,      11,       13, ...
  1,   10,    29,     58,      97,     146,      205, ...
  1,   36,   185,    532,    1161,    2156,     3601, ...
  1,  137,  1257,   5209,   14849,   34041,    67657, ...
  1,  543,  8925,  53347,  198729,  562551,  1330693, ...
  1, 2219, 65445, 564499, 2748641, 9608811, 27053749, ...

Columns k=0..4 give A000012, A002212(n+1), A127846(n+1), A386362, A386389.
Main diagonal gives A386432.
Cf. A000108, A340968.

a(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*(j+1))!/((j+1)!*(j+2)!));

G.f. of column k: (1/x) * Series_Reversion( x/(1+(2*k+1)*x+(k*x)^2) ).
G.f. of column k: 2/(1 - (2*k+1)*x + sqrt((1-x) * (1-(4*k+1)*x))).
A(n,k) = (A340968(n+1,k) - A340968(n,k))/k for k > 0.
(n+2)*A(n,k) = (2*k+1)*(2*n+1)*A(n-1,k) - (4*k+1)*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * (2*k+1)^(n-2*j) * binomial(n,2*j) * Catalan(j).

cp's OEIS Frontend

A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

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A182122 Expansion of exp( arcsinh( 2*x ) ).

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A386362 Expansion of (1/x) * Series_Reversion( x/(1+7*x+9*x^2) ).

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A386389 Expansion of (1/x) * Series_Reversion( x/(1+9*x+16*x^2) ).

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A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

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A127847 a(n)=4^C(n,2)*(4^n-1)/3.

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A386362 Expansion of (1/x) * Series_Reversion( x/(1+7x+9x^2) ).

A386389 Expansion of (1/x) * Series_Reversion( x/(1+9x+16x^2) ).