A092566 -id:A092566 - cp's OEIS frontend

A006442 Expansion of 1/sqrt(1 - 10*x + x^2).

1, 5, 37, 305, 2641, 23525, 213445, 1961825, 18205345, 170195525, 1600472677, 15122515985, 143457011569, 1365435096485, 13033485491077, 124715953657025, 1195966908404545, 11490534389896325, 110584004488276645, 1065853221648055025

Offset: 0

N. J. A. Sloane

Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H can choose from two colors. - Paul Barry, May 25 2005
Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,1), and two kinds of steps (1,0). - Joerg Arndt, Jul 01 2011
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 09 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Column k=2 of A335333.
Sequences of the form LegendreP(n, 2*m+1): A000012 (m=0), A001850 (m=1), this sequence (m=2), A084768 (m=3), A084769 (m=4).
Cf. A098270, A243943 (a(n)^2).

[Evaluate(LegendrePolynomial(n), 5): n in [0..40]]; // G. C. Greubel, May 21 2023

seq(orthopoly[P](n,5), n = 0 .. 20); # Robert Israel, Aug 18 2014

Table[LegendreP[n, 5], {n, 0, 19}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
CoefficientList[Series[1 / Sqrt[1 - 10 x + x^2], {x, 0, 20}], x] (* Vincenzo Librandi, Nov 23 2014 *)

PARI
```
a(n)=subst(pollegendre(n),x,5)
```

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [1,0], [0,1], [1,1]]; /* note the double [1,0] */
/* Joerg Arndt, Jul 01 2011 */

{a(n)=sum(k=0,n,binomial(n,k)^2*2^k*3^(n-k))} /* Paul D. Hanna */

{a(n) = sum(k=0, n, 2^k * binomial(2*k, k) * binomial(n+k, n-k) )}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 17 2014

[gen_legendre_P(n,0,5) for n in range(41)] # G. C. Greubel, May 21 2023

Legendre polynomial evaluated at 5. - Michael Somos, Dec 04 2001
G.f.: 1/sqrt(1 - 10*x + x^2).
a(n) equals the central coefficient of (1 + 5*x + 6*x^2)^n. - Paul D. Hanna, Jun 03 2003
a(n) equals the (n+1)-th term of the binomial transform of 1/(1-2x)^(n+1). - Paul D. Hanna, Sep 29 2003
a(n) = Sum_{k=0..n} 2^k*binomial(n, k)*binomial(n+k, k). - Benoit Cloitre, Apr 13 2004
a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 3^(n-k). - Paul D. Hanna, Feb 04 2012
E.g.f.: exp(5*x) * Bessel_I(0, 2*sqrt(6)*x). - Paul Barry, May 25 2005
D-finite with recurrence: n*a(n) - 5*(2n-1)*a(n-1) + (n-1)*a(n-2) = 0 [Eq (4) in the T. D. Noe article]. R. J. Mathar, Jun 26 2012
a(n) ~ (5 + 2*sqrt(6))^n/(2*sqrt(Pi*n)*sqrt(5*sqrt(6) - 12)). - Vaclav Kotesovec, Oct 05 2012
a(n) = hypergeom([-n, n+1], [1], -2). - Peter Luschny, May 23 2014
a(n) = Sum_{k=0..n} 2^k * C(2*k, k) * C(n+k, n-k). - Paul D. Hanna, Aug 17 2014
a(n) = Sum_{k=0..n} (k+1) * 3^k * (-1)^(n-k) * binomial(n,k) * binomial(n+k+1,n) / (n+k+1). - Vladimir Kruchinin, Nov 23 2014
From Peter Bala, Nov 28 2021: (Start)
a(n) = (1/3)*(1/2)^n*Sum_{k >= n} binomial(k,n)^2*(2/3)^k.
a(n) = (1/3)^(n+1)*hypergeom([n+1, n+1], [1], 2/3).
a(n) = (2^n)*hypergeom([-n, -n], [1], 3/2).
a(n) = [x^n] ((x - 1)*(3 - 2*x))^n
a(n) = (1/2)^n*A098270(n). (End)
a(n) = (-1)^n * Sum_{k=0..n} (1/10)^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 28 2025
a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, Aug 30 2025

A098430 a(n) = 4^n(2n)!/(n!)^2.

Original entry on oeis.org

1, 8, 96, 1280, 17920, 258048, 3784704, 56229888, 843448320, 12745441280, 193730707456, 2958796259328, 45368209309696, 697972450918400, 10768717814169600, 166556168859156480, 2581620617316925440, 40091049586568724480, 623638549124402380800, 9715632133727531827200

Offset: 0

Paul Barry, Sep 07 2004

a(n) counts walks of 2n steps North, East, South or West that start at the origin and end on the line y=x. For example, a(1)=8 counts EW, EN, NE, NS, WE, WS, SN, SW. If the walk has i East and j North steps, then it must have n-j West and n-i South steps. There are Multinomial[i,j,n-j,n-i] ways to arrange these steps and summing over i and j gives the result. - David Callan, Oct 11 2005
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), both of two kinds. - Joerg Arndt, Jul 01 2011
Hankel transform is A121913. - Philippe Deléham, Mar 01 2009
Convolving a(n) with itself yields A001025, the powers of 16. Thus the limiting ratio of this sequence is 16. - Bob Selcoe, Jul 16 2014
Number of strings x of length 4n over the alphabet {1, -1} such that the dot product of x with (x reversed) is 0. - Jeffrey Shallit, Mar 06 2017
Number of orthogonal pairs of vectors of length 2n, constructed with any symmetric binary-valued symbol set. - Ross Drewe, May 18 2018
Diagonal of the rational function 1 / (1 - 4*x - y). - Ilya Gutkovskiy, Apr 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A000302, A000984, A174301, A249308.

a098430 n = a000302 n * a000984 n -- Reinhard Zumkeller, Nov 14 2014

[4^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

A098430 := n -> 4^n*binomial(2*n,n): seq(A098430(n), n=0..30); # Wesley Ivan Hurt, Jul 16 2014

CoefficientList[Series[1/Sqrt[1 - 16 x], {x, 0, 16}], x] (* Robert G. Wilson v, Jun 28 2012 *)
Table[4^n(2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Aug 13 2021 *)

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [1,0], [0,1], [0,1]]; /* note the double [1,0] and [0,1] */
/* Joerg Arndt, Jul 01 2011 */

a = lambda n: 16^n*hypergeometric([-2*n, 1/2], [1], 2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

a(n) = 4^n*binomial(2*n, n) = 4^n*A000984(n).
E.g.f.: exp(8*x)*BesselI(0, 8*x).
G.f.: 1/sqrt(1-16*x). - Zerinvary Lajos, Dec 20 2008, corrected R. J. Mathar, May 18 2009
a(n) = (1/Pi)*Integral_{x=-2..2} (2*x)^(2*n)/sqrt((2-x)*(2+x)) dx. - Peter Luschny, Sep 12 2011
D-finite with recurrence: n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 10 2014
a(n) = A249308(2*n). - Reinhard Zumkeller, Nov 14 2014
a(n) = 16^n*hypergeometric([-2*n, 1/2], [1], 2). - Peter Luschny, May 19 2015
a(n) = A174301(2n,n). - Alois P. Heinz, Apr 15 2019
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 16/15 + 16*sqrt(15)*arcsin(1/4)/225.
Sum_{n>=0} (-1)^n/a(n) = 16/17 - 16*sqrt(17)*arcsinh(1/4)/289. (End)
a(n) = Sum_{k = 0..2*n} (-1)^k *A000984(k) * A000984(2*n-k). Cf. Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k) = 16^n. - Peter Bala, Aug 23 2025

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

A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

Original entry on oeis.org

1, 3, 21, 157, 1239, 10047, 82951, 693603, 5854581, 49778997, 425712429, 3657968097, 31555053921, 273109567797, 2370474720369, 20625186298269, 179841473895447, 1571088267426447, 13747953837604959, 120482775658910763, 1057293764707074027, 9289536349244758791, 81709329486947791419

Offset: 0

Eric Werley, Jun 29 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A091533.

FullSimplify[CoefficientList[Series[(3-6*x+Sqrt[-1+4*x*(9*x-11)+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1]])/(Sqrt[10+8*x]*Sqrt[(1-x)*(1-9*x)]*(4*x*(9*x-11)-1+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1])^(1/4)), {x, 0, 10}], x]]

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [1,1]];
/* Joerg Arndt, Jun 30 2011 */

From Vaclav Kotesovec, Oct 24 2012: (Start)
G.f.: (3 - 6*x + sqrt(-1 + 4*x*(9*x-11) + 4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))) / (sqrt(10+8*x)*sqrt((1-x)*(1-9*x))*(4*x*(9*x-11)-1+4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))^(1/4))
D-finite with recurrence: 15*(n-1)*n*a(n) = (n-1)*(133*n-54)*a(n-1) + (31*n^2 - 177*n + 224)*a(n-2) - (113*n^2 - 295*n + 144)*a(n-3) - 18*(n-3)*(2*n-5)*a(n-4)
a(n) ~ 3^(2*n+3/2)/(2*sqrt(14*Pi*n))
(End)
a(n) = A091533(2*n,n) for n >= 0. - Paul D. Hanna, Dec 11 2018
a(n) = [x^n*y^n] 1/(1 - x - y - x^2 - x*y - y^2) for n >= 0. - Paul D. Hanna, Dec 11 2018

Terms > 425712429 by Joerg Arndt, Jun 30 2011

A137644 a(n) = Sum_{k=0..n} C(n+k,k)*C(n+k,n-k).

Original entry on oeis.org

1, 3, 16, 95, 591, 3780, 24620, 162423, 1081780, 7258053, 48982176, 332140328, 2261099491, 15444137880, 105789736896, 726426836103, 4998885106599, 34464824536500, 238017084356680, 1646234203000485, 11401464090042224, 79060352485691272, 548829398923188036

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,1), (0,2). - Eric Werley, Jun 29 2011

Diagonal of rational function 1/(1 - (x + y + x*y + x^2)). - Gheorghe Coserea, Aug 31 2018

			The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (0,1), (0,2) begins:
  1;
  1,  3;
  1,  5,  16;
  1,  7,  29,  95;
  1,  9,  46, 179,  591;
  1, 11,  67, 303, 1140,  3780;
  1, 13,  92, 475, 2010,  7405, 24620;
  1, 15, 121, 703, 3309, 13427, 48761, 162423;
  1, 17, 154, 995, 5161, 22892, 90241, 324317, 1081780;
This sequence is the diagonal. - _Joerg Arndt_, Jul 01 2011

Indranil Ghosh, Table of n, a(n) for n = 0..500

Cf. A063967.

Table[ HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1/2, 1}, -(1/4)], {n,0,20}] (* Olivier Gérard, Apr 23 2009 *)
Table[Sum[Binomial[n+k,k]Binomial[n+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Aug 03 2011 *)

a(n)=sum(k=0,n,binomial(n+k,k)*binomial(n+k,n-k))

/* same as in A092566 but use */
steps=[[1,0], [1,1], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */

a(n) = 3F2( {-n, n+1, n+1}; {1/2, 1})( -(1/4) ). - Olivier Gérard, Apr 23 2009
G.f.: F'(x)/(1+F(x)), where F(x)=x*(1+F(x))/(1-F(x)-F(x)^2). - Vladimir Kruchinin, Mar 24 2012
a(n) = A063967(n,n). - Alois P. Heinz, Oct 11 2017
a(n) ~ sqrt(56 + (7*(15953 - 267*sqrt(105)))^(1/3) + (7*(15953 + 267*sqrt(105)))^(1/3)) * (((36 + (44766 - 1050*sqrt(105))^(1/3) + (6*(7461 + 175*sqrt(105)))^(1/3))/15)^n / sqrt(210*Pi*n)). - Vaclav Kotesovec, Feb 17 2024

A038112 a(n) = T(2n,n), where T(n,k) is in A037027.

Original entry on oeis.org

1, 2, 9, 40, 190, 924, 4578, 22968, 116325, 593450, 3045185, 15699840, 81260816, 421993040, 2197653240, 11472991008, 60023749566, 314621200260, 1651883008050, 8685998428800, 45734484854520, 241098942106440, 1272406536645660

Offset: 0

Floor van Lamoen

Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011

Diagonal of rational function 1/(1 - (x + y + y^2)). - Gheorghe Coserea, Aug 06 2018

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3 example 7.

List([0..25],n->Sum([0..n],k->Binomial(n+k,k)*Binomial(k,n-k))); # Muniru A Asiru, Aug 06 2018

a:=n->sum(binomial(2*j,n)*(binomial(n+j,2*j)),j=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, Aug 22 2006
series(RootOf((x+1)*(27*x-5)*A^3+4*A+1,A),x=0,30); # Mark van Hoeij, May 01 2013

Table[Sum[Binomial[n+k,k]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Sep 30 2012 *)
Table[Binomial[2 n, n] Hypergeometric2F1[1/2 - n/2, -n/2, -2 n, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *)

{a(n) = if( n<0, 0, sum(k=0, n\2, (2*n-k)!/ (k! * (n-2*k)!)) / n!)}; /* Michael Somos, Sep 29 2003 */

{a(n) = if( n<0, 0, n++; n * polcoeff(serreverse( x - x^2 - x^3 + x * O(x^n)), n))}; /* Michael Somos, Sep 29 2003 */

/* same as in A092566 but use */
steps=[[0,1], [1,0], [2,0]]; /* Joerg Arndt, Jun 30 2011 */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^m/m!)); polcoeff(A, n)} \\ Paul D. Hanna, Aug 04 2012

a(n) = Sum_{k=0..n} C(n+k,k)*C(k,n-k). - Paul Barry, May 13 2006
a(n) = Sum_{j=0..n} binomial(2*j, n)*binomial(n+j, 2*j). - Zerinvary Lajos, Aug 22 2006
a(n) = [x^n] (1/(1-x-x^2))^(n+1). - Paul Barry, Mar 23 2011
a(n) = (n+1)*A001002(n+1).
G.f.: Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n!. - Paul D. Hanna, Aug 04 2012
Recurrence: 5*(n-1)*n*a(n) = 11*(n-1)*(2*n-1)*a(n-1) + 3*(3*n-4)*(3*n-2)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3^(3*n+3/2)/(2^(3/2)*5^(n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: A(x) where (x+1)*(27*x-5)*A(x)^3 + 4*A(x) + 1 = 0. - Mark van Hoeij, May 01 2013

A098658 a(n) = 3^n(2n)!/(n!)^2.

Original entry on oeis.org

1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070, 956987460, 10909657044, 124965162504, 1437099368796, 16581915793800, 191876454185400, 2225766868550640, 25874539846901190, 301362287628613860

Offset: 0

Paul Barry, Sep 20 2004

Number of lattice paths from (0,0) to (n,n) using steps (0,1) and three kinds of steps (1,0). - Joerg Arndt, Jul 01 2011
Sixth binomial transform of 1/sqrt(1-36*x^2).
Diagonal of the rational function 1 / (1 - 3*x - y). - Ilya Gutkovskiy, Apr 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A000984, A059304, A101600, A322246.

[3^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

Table[3^n (2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Dec 14 2011 *)

/* same as in A092566 but use */
steps=[[1,0], [1,0], [1,0], [0,1]]; /* note the triple [1,0] */
/* Joerg Arndt, Jun 30 2011 */

G.f.: 1/sqrt((1-6*x)^2-36*x^2) = 1/sqrt(1-12*x).
E.g.f.: exp(6*x)*BesselI(0, 6x).
a(n) = [t^n](1+6*t+9*t^2)^n.
a(n) = 3^n*A000984(n). - R. J. Mathar, Oct 10 2012
G.f.: Q(0), where Q(k) = 1 + 12*x*(4*k+1)/( 4*k+2 - 12*x*(4*k+2)*(4*k+3)/(12*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
n*a(n) +6*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 27 2014
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 45*x^2 + 378*x^3 + ... is the o.g.f. for A101600. - Peter Bala, Jul 16 2015
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 12/11 + 12*sqrt(11)*arcsin(1/sqrt(12))/121.
Sum_{n>=0} (-1)^n/a(n) = 12/13 - 12*sqrt(13)*arcsinh(1/sqrt(12))/169. (End)
From Peter Bala, Oct 12 2024: (Start)
a(n) = Integral_{x = 0..12} x^n * w(x) dx, where w(x) = 1/( Pi*sqrt(x*(12 - x)) ) is positive on the interval (0, 12). The weight function w(x) is singular at x = 0 and at x = 12 and is the solution of the Hausdorff moment problem.
Binomial transform of A322246.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r. (End)

A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).

Original entry on oeis.org

1, 3, 22, 165, 1327, 10950, 92045, 783579, 6733966, 58294401, 507579829, 4440544722, 39000863629, 343677908223, 3037104558574, 26904952725061, 238854984979423, 2124492829796598, 18927927904130617, 168888613467092895, 1508973226894216106, 13498652154574126523, 120886709687492946083

Offset: 0

Eric Werley, Jun 29 2011

nonn

Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x*y - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

p4 := x^4+6*x^3+7*x^2-10*x+1;
ogf := sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) );
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 16 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, 1, add(b(x, y-j)+
         b(x-j, y)+b(x-j, y-j), j=1..2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2017

b[x_, y_] := b[x, y] = If[Min[x, y] < 0, 0, If[Max[x, y] == 0, 1, Sum[b[x, y - j] + b[x - j, y] + b[x - j, y - j], {j, 1, 2}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [1,1], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) ) where p4 = x^4+6*x^3+7*x^2-10*x+1. - Mark van Hoeij, Apr 16 2013

Terms > 507579829 from Joerg Arndt, Jun 30 2011

A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.

Original entry on oeis.org

1, 1, 4, 9, 29, 82, 255, 773, 2410, 7499, 23575, 74298, 235325, 747407, 2381126, 7603433, 24332595, 78013192, 250540055, 805803691, 2595158718, 8368026845, 27012184877, 87283372610, 282294378071, 913775677281, 2960160734818

Offset: 0

N. J. A. Sloane, Jan 27 2001

Number of paths in the right half-plane from (0,0) to (n,0) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=9 because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The number of such paths restricted to the first quadrant is given in A128720. - Emeric Deutsch, Sep 03 2007
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (2,2). - Joerg Arndt, Jun 30 2011
Other two columns of the triangle in A059317 are given in A106053 and A106050. - Emeric Deutsch, Sep 03 2007

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

T. D. Noe, Table of n, a(n) for n=0..200
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.

Cf. A128720, A106050, A106053.

Cf. A181545. - Paul D. Hanna, Oct 29 2010

r:=proc(i,j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1,j)+r(i-1,j-1)+r(i-1,j+1)+r(i-2,j) else 0 fi end: seq(r(i,0),i=1..12); # very slow; Emeric Deutsch, Jun 06 2004
G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z, n),n=0..27); # Emeric Deutsch, Sep 03 2007
a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-1)*a[n-3])/(n+1) end do: seq(a[n],n=0..27); # Emeric Deutsch, Sep 03 2007

CoefficientList[Series[1/Sqrt[(1+x-x^2)(1-3x-x^2)],{x,0,40}],x] (* Harvey P. Dale, Jun 04 2011 *)
a[n_] := Sum[Binomial[n-k, k]*Hypergeometric2F1[(2*k-n)/2, (2*k-n+1)/2, 1, 4], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 26 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^(2*m)/(1-x-x^2+x*O(x^n))^(2*m+1)*(2*m)!/(m!)^2),n)} \\ Paul D. Hanna, Oct 29 2010

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,2], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: 1/sqrt((1+z-z^2)*(1-3*z-z^2)). - Emeric Deutsch, Sep 03 2007
D-finite with recurrence: (n+1)*a(n+1)=(2*n+1)*a(n)+5*n*a(n-1)-(2*n-1)*a(n-2)-(n-1)*a(n-3). - Emeric Deutsch, Sep 03 2007
a(n) = sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}. - Paul Barry, Nov 29 2008
G.f.: A(x) = Sum_{n>=0} (2*n)!/(n!)^2 * x^(2n)/(1-x-x^2)^(2n+1). - Paul D. Hanna, Oct 29 2010
a(n) ~ sqrt((3+11/sqrt(13))/8) * ((3+sqrt(13))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 11 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001

A192369 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (0,2), (1,0), (2,0), and (2,2).

Original entry on oeis.org

1, 2, 15, 90, 617, 4248, 29945, 213404, 1535661, 11129314, 81123369, 594092166, 4367701295, 32216566492, 238301617605, 1766979857196, 13129849298327, 97746629874786, 728897653778335, 5443488765350770, 40706993579981847, 304779612155116444, 2284440756129389775, 17139937071103287600

Offset: 0

Eric Werley, Jun 29 2011

nonn

Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

p4 := (x-1)*(x^3+5*x^2+7*x-1);
ogf := sqrt(((2*x^2+4*x-3)/p4-2/sqrt(p4))/(4*x^2-8*x-5));
series(ogf, x=0, 30); # Mark van Hoeij, Apr 16 2013

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f. is a nested square root, see Maple program. - Mark van Hoeij, Apr 16 2013

Terms > 81123369 from Joerg Arndt, Jun 30 2011

cp's OEIS Frontend

A006442 Expansion of 1/sqrt(1 - 10*x + x^2).

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A098430 a(n) = 4^n*(2*n)!/(n!)^2.

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

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A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

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A137644 a(n) = Sum_{k=0..n} C(n+k,k)*C(n+k,n-k).

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A038112 a(n) = T(2n,n), where T(n,k) is in A037027.

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A098430 a(n) = 4^n(2n)!/(n!)^2.

A098658 a(n) = 3^n(2n)!/(n!)^2.