A137635 -id:A137635 - cp's OEIS frontend

A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

1, 2, 14, 80, 486, 3030, 19184, 122924, 794678, 5173160, 33863666, 222683588, 1469908848, 9733916596, 64636957300, 430240178484, 2869778018070, 19177245746844, 128361805431752, 860443079597872, 5775392952659170, 38811408514848032, 261101034656317244

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361814.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^4))

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(4*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) + 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

Original entry on oeis.org

1, 2, 16, 100, 660, 4482, 30886, 215364, 1515000, 10730800, 76426846, 546792056, 3926775646, 28290272420, 204375145480, 1479963148220, 10739326203132, 78072933869364, 568503202324540, 4145718464390120, 30271771382355430, 221305746414518180

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1139

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361813.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^5))

a(n)= sum(k=0, n, binomial(2*k,k) * binomial(5*k,n-k)) \\ Winston de Greef, Mar 25 2023

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(5*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 5*(2*n-2)*a(n-2) + 10*(2*n-3)*a(n-3) + 10*(2*n-4)*a(n-4) + 5*(2*n-5)*a(n-5) + (2*n-6)*a(n-6) ) for n > 5.

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    2,   2,    2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   46,   62,   80,  100, ...
   70, 136,  226,  342,  486,  660, ...
  252, 592, 1136, 1932, 3030, 4482, ...

Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Cf. A099233, A361834.

T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));

G.f. of column k: 1/sqrt(1 - 4*x*(1+x)^k).

n*T(n,k) = 2 * Sum_{j=0..k} binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A361841 Expansion of 1/(1 - 9x(1+x)^2)^(1/3).

Original entry on oeis.org

1, 3, 24, 201, 1809, 16893, 161676, 1574289, 15527052, 154662930, 1552725504, 15688410264, 159355067283, 1625899880673, 16652520666414, 171119405299005, 1763475423260049, 18219685282559559, 188664151412242368, 1957539823296458841, 20347733657193596127

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..966

Column k=2 of A361839.

Cf. A002478, A137635, A361844.

A361841 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], -3/4):
seq(simplify(A361841(n)), n = 0..20); # Peter Luschny, Mar 27 2023

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^2)^(1/3))

n*a(n) = 3 * ( (3*n-2)*a(n-1) + 2*(3*n-4)*a(n-2) + (3*n-6)*a(n-3) ) for n > 2.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(2*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], -3/4). - Peter Luschny, Mar 27 2023

A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).

Original entry on oeis.org

1, 3, 17, 101, 627, 3999, 25955, 170571, 1131433, 7559301, 50795985, 342935689, 2324278669, 15804931797, 107775401349, 736723618773, 5046774983235, 34636814325087, 238114193665451, 1639378334244867, 11301978856210543, 78010917772099207, 539055832175992119

Offset: 0

Kent Mei, Dec 08 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1179 (first 101 terms from Kent Mei)

Cf. A000984, A001850, A137635, A339390, A382436.

a:= proc(n) local t; 1/(1-x-y-x*y-(x*y^2)-(x^2*y));
      for t in [x, y] do coeftayl(%, t=0, n) od
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# second Maple program:
b:= proc(l) option remember; `if`(l[2]=0, 1,
      add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
      [[1, 0], [0, 1], [1$2], [1, 2], [2, 1]]))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# third Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 3, 17][n+1],
      ((6*n-3)*a(n-1)+(7*n-7)*a(n-2)+(4*n-6)*a(n-3))/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020

b[l_] := b[l] = If[l[[2]] == 0, 1,
     Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l - h]], {h,
     {{1, 0}, {0, 1}, {1, 1}, {1, 2}, {2, 1}}}]];
a[n_] := b[{n, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

a(n) = [(x*y)^n] 1/(1-x-y-x*y-x*y^2-x^2*y). - Alois P. Heinz, Dec 09 2020
a(n) = A382436(2n,n). - Alois P. Heinz, Mar 25 2025
a(n) ~ sqrt((3776 + (26570110976 - 74946048*sqrt(177))^(1/3) + 8*(59*(879572 + 2481*sqrt(177)))^(1/3))/11328) * (2 + (459 - 12*sqrt(177))^(1/3)/3 + (153 + 4*sqrt(177))^(1/3)/3^(2/3))^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 26 2025

A361726 Diagonal of rational function 1/(1 - (1 + xy) (x^2 + y^2)).

Original entry on oeis.org

1, 0, 2, 4, 8, 24, 56, 144, 376, 960, 2512, 6560, 17184, 45248, 119296, 315392, 835552, 2217216, 5893568, 15687552, 41810944, 111567104, 298016512, 796832256, 2132456704, 5711486976, 15309014528, 41062927360, 110213725184, 295995574272, 795391639552

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

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

Cf. A137635, A361727.

Cf. A115962, A360266.

a(n) = sum(k=0, n\2, binomial(2*k, k)*binomial(2*k, n-2*k));

G.f.: 1/sqrt(1 - 4 * x^2 * (1+x)^2).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k,k) * binomial(2*k,n-2*k).
a(n) ~ (1 + sqrt(3))^(n + 1/2) / (2*3^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 22 2023
n*a(n) = 2*(2*n-2)*a(n-2) + 4*(2*n-3)*a(n-3) + 2*(2*n-4)*a(n-4) for n > 3. - Seiichi Manyama, Mar 23 2023

A361727 Diagonal of rational function 1/(1 - (1 + xy) (x^3 + y^3)).

Original entry on oeis.org

1, 0, 0, 2, 4, 2, 6, 24, 36, 44, 126, 300, 470, 860, 2080, 4192, 7420, 15260, 33124, 64568, 124558, 259632, 535668, 1055460, 2118414, 4373412, 8872644, 17765396, 36138168, 73972404, 149793424, 303140552, 618565948, 1261454064, 2561056212, 5211145368

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..3162

Cf. A137635, A361726.

a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(2*k, n-3*k));

G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)^2).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(2*k,n-3*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 8*(n-2)*a(n-4) + 2*(2*n-5)*a(n-5).
a(n) ~ 1 / (sqrt((5 - 8*r^3 - 8*r^4)*Pi*n) * r^n), where r = 0.484163615233802299545617907511361266999078019358842974840776720... is the real root of the equation -1 + 4*r^3 + 8*r^4 + 4*r^5 = 0. (End)

A377194 Expansion of 1/(1 - 4x(1+x)^2)^(3/2).

Original entry on oeis.org

1, 6, 42, 266, 1650, 10032, 60202, 357744, 2109882, 12369280, 72163560, 419315340, 2428226530, 14021002860, 80757350040, 464127134636, 2662303793226, 15245389224492, 87168383093576, 497721319382220, 2838427001118456, 16168991846946656, 92012074475132892

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

Cf. A137635, A377195, A377196.

a[n_]:=Sum[(2*k+1)Binomial[2*k,k]Binomial[2k,n-k],{k,0,n}]; Array[a,23,0] (* Stefano Spezia, May 08 2025 *)

a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*k, n-k));

a(0) = 1, a(1) = 6, a(2) = 42; a(n) = (2*(2*n+1)*a(n-1) + 8*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.

a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*k,n-k).

A361752 a(n) = Sum_{k=0..floor(n/2)} binomial(2(n-2k),k) * binomial(2(n-2k),n-2*k).

Original entry on oeis.org

1, 2, 6, 24, 94, 374, 1520, 6252, 25942, 108408, 455586, 1923444, 8151856, 34661252, 147788484, 631660788, 2705471254, 11609393084, 49899207640, 214792704256, 925811868178, 3995288307392, 17260287754284, 74641620619072, 323080683587056, 1399606566298916

Offset: 0

Seiichi Manyama, Mar 23 2023

nonn

Diagonal of rational function 1/(1 - (1 + (x*y)^2) * (x + y)).

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

Cf. A137635, A361753.

Cf. A360266.

a(n) = sum(k=0, n\2, binomial(2*(n-2*k), k)*binomial(2*(n-2*k), n-2*k));

from math import comb
def A361752(n): return sum(comb(m:=(r:=n-(k<<1))<<1,k)*comb(m,r) for k in range((n>>1)+1)) # Chai Wah Wu, Mar 23 2023

G.f.: 1/sqrt(1 - 4*x*(1 + x^2)^2).

n*a(n) = 2*(2*n-1)*a(n-1) + 4*(2*n-3)*a(n-3) + 2*(2*n-5)*a(n-5) for n > 4.

A361753 a(n) = Sum_{k=0..floor(n/3)} binomial(2(n-3k),k) * binomial(2(n-3k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 74, 276, 1044, 3994, 15426, 60008, 234764, 922716, 3640700, 14411952, 57210750, 227659704, 907853778, 3627085932, 14515139376, 58174092472, 233463067284, 938061587212, 3773298437204, 15193083455580, 61230698571372, 246978403761112

Offset: 0

Seiichi Manyama, Mar 23 2023

nonn

Diagonal of rational function 1/(1 - (1 + (x*y)^3) * (x + y)).

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

Cf. A137635, A361752.

Cf. A360267.

a(n) = sum(k=0, n\3, binomial(2*(n-3*k), k)*binomial(2*(n-3*k), n-3*k));

from math import comb
def A361753(n): return sum(comb(m:=(r:=n-3*k)<<1,k)*comb(m,r) for k in range(n//3+1)) # Chai Wah Wu, Mar 23 2023

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)^2).

Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 8*(n-2)*a(n-4) + 2*(2*n-7)*a(n-7). - Vaclav Kotesovec, Mar 23 2023

cp's OEIS Frontend

A361813 Expansion of 1/sqrt(1 - 4*x*(1+x)^4).

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A361814 Expansion of 1/sqrt(1 - 4*x*(1+x)^5).

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A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).

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A361841 Expansion of 1/(1 - 9*x*(1+x)^2)^(1/3).

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A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).

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A361726 Diagonal of rational function 1/(1 - (1 + x*y) * (x^2 + y^2)).

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A361727 Diagonal of rational function 1/(1 - (1 + x*y) * (x^3 + y^3)).

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A377194 Expansion of 1/(1 - 4*x*(1+x)^2)^(3/2).

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Mathematica

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A361752 a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k),k) * binomial(2*(n-2*k),n-2*k).

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A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

A361841 Expansion of 1/(1 - 9x(1+x)^2)^(1/3).

A361726 Diagonal of rational function 1/(1 - (1 + xy) (x^2 + y^2)).

A361727 Diagonal of rational function 1/(1 - (1 + xy) (x^3 + y^3)).

A377194 Expansion of 1/(1 - 4x(1+x)^2)^(3/2).

A361752 a(n) = Sum_{k=0..floor(n/2)} binomial(2(n-2k),k) * binomial(2(n-2k),n-2*k).

A361753 a(n) = Sum_{k=0..floor(n/3)} binomial(2(n-3k),k) * binomial(2(n-3k),n-3*k).