A006139 -id:A006139 - cp's OEIS frontend

A374513 Expansion of 1/(1 - 4x - 4x^2)^(7/2).

1, 14, 140, 1176, 8904, 62832, 421344, 2718144, 17008992, 103847744, 621292672, 3654187264, 21182563584, 121263109632, 686660004864, 3851149940736, 21416533501440, 118199459288064, 647926485764096, 3529938203545600, 19124354344775680

Offset: 0

Seiichi Manyama, Jul 09 2024

nonn

Cf. A006139, A374497, A374511.

a[n_]:=2^(n-4) Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, 2]/45; Array[a,21,0] (* Stefano Spezia, Jul 10 2024 *)

a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, 2^(n-k)*binomial(n+3, n-2*k)*binomial(2*k+3, k));

a(0) = 1, a(1) = 14; a(n) = (2*(2*n+5)*a(n-1) + 4*(n+5)*a(n-2))/n.
a(n) = (binomial(n+6,3)/20) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = 2^(n-4)*Pochhammer(n+1, 6)*hypergeom([(1-n)/2, -n/2], [4], 2)/45. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

A260774 Certain directed lattice paths.

Original entry on oeis.org

1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Alois P. Heinz, Table of n, a(n) for n = 0..1235 (first 101 terms from Lars Blomberg)
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Cf. A122868, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021

b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
     If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
     If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

See Dziemianczuk (2014) Equation (33a) with m=1.
From Vaclav Kotesovec, Jul 15 2022: (Start)
Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).
a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)

More terms from Lars Blomberg, Aug 01 2015

A102051 Matrix inverse of triangle A101275 (number of Schröder paths).

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -9, 15, -7, 1, 31, -58, 36, -10, 1, -113, 229, -170, 66, -13, 1, 431, -924, 775, -372, 105, -16, 1, -1697, 3795, -3481, 1939, -691, 153, -19, 1, 6847, -15822, 15542, -9674, 4072, -1154, 210, -22, 1, -28161, 66801, -69276, 47012, -22446, 7606, -1788, 276, -25, 1

Offset: 0

Paul D. Hanna, Dec 27 2004

Row sums are {1,0,0,0...}. Absolute row sums form A006139. Column 0 forms signed A052709. Column 1 forms A102052. Column 2 forms A102053.

			Rows begin:
[1],
[ -1,1],
[3,-4,1],
[ -9,15,-7,1],
[31,-58,36,-10,1],
[ -113,229,-170,66,-13,1],
[431,-924,775,-372,105,-16,1],
[ -1697,3795,-3481,1939,-691,153,-19,1],
[6847,-15822,15542,-9674,4072,-1154,210,-22,1],...
Matrix inverse equals triangle A101275:
[1],
[1,1],
[1,4,1],
[1,13,7,1],
[1,44,34,10,1],...

Cf. A101275, A006139, A052709, A102052, A102053, A039599.

T(n,m):=(-1)^(n-m)*(2*m+1)*(sum((binomial(k,n-k)*binomial(2*k,k-m))/(m+k+1),k,0,n)); /* Vladimir Kruchinin, Apr 18 2015 */

{T(n,k)=polcoeff(polcoeff(2/(2*y+(1-y)*(1+sqrt(1+4*x-4*x^2+x*O(x^n)))),n)+y*O(y^k),k)}

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

T(n,m) = (-1)^(n-m)*(2*m+1)*Sum_{k=0..n} C(k,n-k)*C(2*k,k-m)/(m+k+1). - Vladimir Kruchinin, Apr 18 2015

A110135 Square array of expansions of 1/sqrt(1-4x-4kx^2), read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 20, 8, 2, 1, 70, 32, 10, 2, 1, 252, 136, 44, 12, 2, 1, 924, 592, 214, 56, 14, 2, 1, 3432, 2624, 1052, 304, 68, 16, 2, 1, 12870, 11776, 5284, 1632, 406, 80, 18, 2, 1, 48620, 53344, 26840, 9024, 2332, 520, 92, 20, 2, 1, 184756, 243392, 137638, 50304

Offset: 0

Paul Barry, Jul 13 2005

Column k has g.f. 1/sqrt(1-4x-4*k*x^2) and e.g.f. exp(2x)BesselI(0,2*sqrt(k)x). Columns include A000984, A006139, A084609, A098453. Row sums of triangle are A110136. Diagonal sums of triangle are A110137.

			As a square array, rows start
    1,   1,    1,    1,    1, ...
    2,   2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   44,   56,   68,   80, ...
   70, 136,  214,  304,  406,  520, ...
  252, 592, 1052, 1632, 2332, 3152, ...
As a number triangle, rows start
    1;
    2,   1;
    6,   2,   1;
   20,   8,   2,   1;
   70,  30,  10,   2,   1;
  252, 136,  44,  12,   2,   1;

Square array T(n, k) = Sum_{j=0..floor(n/2)} C(n, j)*C(2(n-j), n)*k^j.

Number triangle T1(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(2(n-k-j), n-k)*k^j;

A110446 Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 32, 24, 6, 1, 136, 128, 48, 8, 1, 592, 680, 320, 80, 10, 1, 2624, 3552, 2040, 640, 120, 12, 1, 11776, 18368, 12432, 4760, 1120, 168, 14, 1, 53344, 94208, 73472, 33152, 9520, 1792, 224, 16, 1, 243392, 480096, 423936, 220416, 74592, 17136

Offset: 0

David Callan, Jul 20 2005

T(n,k) = number of Delannoy paths (A001850) of steps east(E), north(N) and diagonal (D) (i.e., northeast) from (0,0) to (n,n) containing k Ds not preceded by an E.

			Table begins
\ k...0....1....2....3....4....
n\
0 |...1
1 |...2....1
2 |...8....4....1
3 |..32...24....6....1
4 |.136..128...48....8....1
5 |.592..680..320...80...10....1
The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,
and so T(3,2)=6.

Column k=0 is A006139.

T[n_, k_] := SeriesCoefficient[(1-z(4 + 2*t) - z^2 (4 - 4*t - t^2))^(-1/2), {z, 0, n}, {t, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 08 2016 *)

G.f. G(z, t)=Sum_{n>=k>=0}T(n, k)*z^n*t^k is given by G(z, t)= (1 - z(4 + 2*t) - z^2(4 - 4*t - t^2))^(-1/2)

A260772 Certain directed lattice paths.

Original entry on oeis.org

1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.

Cf. A122868, A260774, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

# A260772 satisfies a 4th-order recurrence that can be reduced
# to a 2nd-order recurrence given in this program t:
t := proc(n) options remember;
if n <= 1 then
    [-1/2, 0, 1, 4][2*n+2]
  else
    (16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1))
       /( n*(2*n+1)*(5*n-7) )
  fi
end:
A260772 := proc(n)
t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n)
end:
seq(A260772(i),i=0..100);
# Mark van Hoeij, Jul 14 2022

a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */

a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019

G.f.: P1(x) = (2*(1-x)/3)/x - ((2*sqrt(1-5*x-2*x^2)/3)/x)*sin((Pi/6 + arccos(((20*x^3-6*x^2+15*x-2)/2)/(1-5*x-2*x^2)^(3/2))/3)). - See Dziemianczuk (2014), Proposition 11.
a(n) = (1/n)*Sum_{j=0..(n+1)/4} (-1)^j*C(n,j)*C(3*n-4*j,n-4*j+1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
n*(n+1)*(25*n^2-70*n+21)*a(n) - 30*(7*n-15)*n*a(n-1) + (-1100*n^4+5280*n^3-6424*n^2-1188*n+3816)*a(n-2) + 120*(n+2)*(n-3)*a(n-3) - 16*(n-3)*(n-4)*(25*n^2-20*n-24)*a(n-4) = 0. - Mark van Hoeij, Jul 14 2022
a(n) ~ 2^(n - 1/2) * phi^((10*n - 1)/4) / (sqrt(Pi) * 5^(1/4) * sqrt(phi^(3/2) - 2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 15 2022

More terms from Lars Blomberg, Aug 01 2015

A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 4, 18, 80, 360, 1632, 7448, 34176, 157536, 728960, 3384128, 15754752, 73525504, 343870464, 1611288960, 7562801152, 35550504448, 167339022336, 788643765248, 3720901222400, 17573439614976, 83074892775424, 393056192851968, 1861155016212480, 8819174122700800, 41818448615636992

Offset: 0

Seiichi Manyama, Aug 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A006139, A387402, A387403.

Cf. A071356, A374497.

[&+[2^(n-k) * Binomial(n+1,n-2*k) * Binomial(2*k+1,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Table[Sum[2^(n-k)*Binomial[n+1,n-2*k]*Binomial[2*k+1,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)

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

n*(n+2)*a(n) = (n+1) * (2*(2*n+1)*a(n-1) + 4*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+1).
E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.
a(n) = (n+1) * A071356(n).

A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 6, 32, 160, 780, 3752, 17920, 85248, 404640, 1918400, 9090048, 43064320, 204032192, 966887040, 4583424000, 21735350272, 103114538496, 489392157696, 2323701678080, 11037970513920, 52454251902976, 249373626208256, 1186024281341952, 5642924625100800, 26858183388774400, 127880625111662592

Offset: 0

Seiichi Manyama, Aug 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A006139, A387401, A387403.

Cf. A374511.

[&+[2^(n-k) * Binomial(n+2,n-2*k) * Binomial(2*k+2,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Table[Sum[2^(n-k)*Binomial[n+2,n-2*k]*Binomial[2*k+2,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)

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

n*(n+4)*a(n) = (n+2) * (2*(2*n+3)*a(n-1) + 4*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+2).
E.g.f.: exp(2*x) * BesselI(2, 2*sqrt(2)*x) / 2, with offset 2.

A387403 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 8, 50, 280, 1484, 7616, 38304, 190080, 934560, 4564736, 22189024, 107476096, 519180480, 2502850560, 12046666752, 57912029184, 278136798720, 1334832967680, 6402435630080, 30695114813440, 147110418036736, 704860523102208, 3376580007936000, 16172904859238400

Offset: 0

Seiichi Manyama, Aug 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A006139, A387401, A387402.

Cf. A374513.

[&+[2^(n-k) * Binomial(n+3,n-2*k) * Binomial(2*k+3,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Table[Sum[2^(n-k)*Binomial[n+3,n-2*k]*Binomial[2*k+3,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)

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

n*(n+6)*a(n) = (n+3) * (2*(2*n+5)*a(n-1) + 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+3).
E.g.f.: exp(2*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 14, 32, 70, 1, 2, 24, 68, 136, 252, 1, 2, 38, 128, 406, 592, 924, 1, 2, 56, 212, 1096, 2332, 2624, 3432, 1, 2, 78, 320, 2566, 7632, 13964, 11776, 12870, 1, 2, 104, 452, 5320, 20092, 60864, 83848, 53344, 48620

Offset: 0

Seiichi Manyama, Aug 29 2025

			Square array begins:
    1,    1,     1,     1,      1,      1,       1, ...
    2,    2,     2,     2,      2,      2,       2, ...
    6,    8,    14,    24,     38,     56,      78, ...
   20,   32,    68,   128,    212,    320,     452, ...
   70,  136,   406,  1096,   2566,   5320,   10006, ...
  252,  592,  2332,  7632,  20092,  44752,   88092, ...
  924, 2624, 13964, 60864, 210524, 607424, 1523724, ...

Columns k=0..4 give A000984, A006139, A084770, A098455, A098456.
Main diagonal gives A387467.
Cf. A386621.

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

A(n,k) = Sum_{j=0..n} (1-k*i)^j * (1+k*i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*(2*n-1)*A(n-1,k) + 4*k^2*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * 2^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*x) * BesselI(0, 2*sqrt(k^2+1)*x).

cp's OEIS Frontend

A374513 Expansion of 1/(1 - 4*x - 4*x^2)^(7/2).

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A260774 Certain directed lattice paths.

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Maple

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A102051 Matrix inverse of triangle A101275 (number of Schröder paths).

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A110135 Square array of expansions of 1/sqrt(1-4x-4*k*x^2), read by antidiagonals.

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A110446 Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.

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Mathematica

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A260772 Certain directed lattice paths.

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Maple

Maxima

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Extensions

A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

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Magma

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A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.

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A374513 Expansion of 1/(1 - 4x - 4x^2)^(7/2).

A110135 Square array of expansions of 1/sqrt(1-4x-4kx^2), read by antidiagonals.

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).