A005566 -id:A005566 - cp's OEIS frontend

A064036 Number of walks of length n on cubic lattice, starting at origin, staying in first (nonnegative) octant.

1, 3, 12, 51, 234, 1110, 5460, 27405, 140490, 729918, 3845016, 20447658, 109801692, 593806356, 3234529584, 17715445605, 97567971930, 539701180590, 2998595422680, 16719506691030, 93559970043540, 525093580540620, 2955822168597480, 16680150247605390, 94365481922990460

Offset: 0

Henry Bottomley, Aug 23 2001

nonn

			a(2)=12 since a(1) is obviously 3 and from each of these three positions there are four possible steps which remain in the first octant.

Robert Israel, Table of n, a(n) for n = 0..500
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

Cf. A064037. The two- and one-dimensional equivalents are A005566 and A001405. With no restriction on the walks, the number is 6^n, i.e. A000400.

S:= series((BesselI(0,2*x)+BesselI(1,2*x))^3, x, 101):
seq(simplify(coeff(S,x,n))*n!, n=0..100); # Robert Israel, Oct 10 2016

a(n) = sum_j[C(n, j)B(j)B(j+1)B(n-j)] where B(k)=C(k, [k/2])=A001405(k)
E.g.f.: (BesselI(0, 2*x)+BesselI(1, 2*x))^3. - Vladeta Jovovic, Apr 28 2003
From Vaclav Kotesovec, Jun 10 2019: (Start)
Recurrence: (n+1)*(n+2)*(n+3)*a(n) = 4*(5*n^2+10*n+3)*a(n-1) + 4*(n-1)*(10*n^2+10*n-9)*a(n-2) - 144*(n-2)*(n-1)*a(n-3) - 144*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 6^(n + 3/2) / (Pi*n)^(3/2). (End)

A360858 Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 18, 1, 5, 20, 40, 60, 1, 6, 30, 75, 150, 200, 1, 7, 42, 126, 315, 525, 700, 1, 8, 56, 196, 588, 1176, 1960, 2450, 1, 9, 72, 288, 1008, 2352, 4704, 7056, 8820, 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752

Offset: 0

Peter Luschny, Feb 28 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 1,  2;
[2] 1,  3,  6;
[3] 1,  4, 12,  18;
[4] 1,  5, 20,  40,   60;
[5] 1,  6, 30,  75,  150,  200;
[6] 1,  7, 42, 126,  315,  525,   700;
[7] 1,  8, 56, 196,  588, 1176,  1960,  2450;
[8] 1,  9, 72, 288, 1008, 2352,  4704,  7056,  8820;
[9] 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752.

Cf. A005566 (main diagonal), A001700 (row sums).

Cf. A360857, A360859.

A360858 := (n, k) -> binomial(n + 1, ceil(k/2))*binomial(n, floor(k/2)):
seq(seq(A360858(n, k), k = 0..n), n = 0..9);

from math import comb
def A360858_T(n,k): return comb(n,m:=k>>1)**2*(n+1)//(m+1 if k&1 else n+1-m) # Chai Wah Wu, Feb 28 2023

A120406 Triangle read by rows: related to series expansion of the square root of 2 linear factors.

Original entry on oeis.org

1, 2, 2, 5, 6, 5, 14, 18, 18, 14, 42, 56, 60, 56, 42, 132, 180, 200, 200, 180, 132, 429, 594, 675, 700, 675, 594, 429, 1430, 2002, 2310, 2450, 2450, 2310, 2002, 1430, 4862, 6864, 8008, 8624, 8820, 8624, 8008, 6864, 4862

Offset: 0

David Callan, Jul 03 2006

The numbers T(n,k) arise in the expansion of the square root of 2 generic linear factors: 1 - sqrt((1-a*x)*(1-b*x)) = (a+b)*x/2 + (1/8)*(b-a)^2*x^2*Sum_{n>=0} (Sum_{k=0..n} T(n,k)*a^k*b^(n-k))*(x/4)^n. (The g.f. below simply reformulates this fact.) A combinatorial interpretation of T(n,k) would be very interesting.

			Table begins
  \ k..0....1....2....3....4....5....6
  n
  0 |..1
  1 |..2....2
  2 |..5....6....5
  3 |.14...18...18...14
  4 |.42...56...60...56...42
  5 |132..180..200..200..180..132
  6 |429..594..675..700..675..594..429

Column k=0 is the Catalan numbers A000108 (offset). The middle-of-row entries form A005566. Cf. A067804.

Table[2 Binomial[n,k]^2 Binomial[2n+2,n]/ Binomial[2n+2,2k+1],{n,0,9},{k,0,n}]

solve(A=x*(A^2*y^2-2*A^2*y-2*A*y+A^2-2*A+1),A); /* Vladimir Kruchinin, Oct 24 2020 */

T(n,k) = 2*binomial(n,k)^2*binomial(2n+2,n)/binomial(2n+2,2k+1). This shows that T(n,k) is positive and the rows are symmetric.
T(n,k) = (k+1)*CatalanNumber(n+1) - 2*Sum_{j=0..k-1} (k-j)*CatalanNumber(j)*CatalanNumber(n-j). This shows that T(n,k) is an integer.
G.f.: F(x,y):=Sum_{n>=0, k=0..n} T(n,k) x^n y^k is given by F(x,y) = ( 1-2x-2x*y-sqrt(1-4x)*sqrt(1-4x*y) )/( 2x^2*(1-y)^2 ). This shows that the row sums are the powers of 4 (A000302) because lim_{y->1} F(x,y) = 1/(1-4x).
1 + x*(d/dx)(log(F(x,y))) = 1 + (2 + 2*y)*x + (6 + 4*y + 6*y^2)*x^2 + ... is the o.g.f. for A067804. - Peter Bala, Jul 17 2015
G.f. A(x,y) = -G(-x,y), G(x,y) satisfies G(x,y) = x/A008459(G(x,y))^2. - Vladimir Kruchinin, Oct 24 2020

A172101 Triangle, read by rows, given by [0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...] DELTA [1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 2, 1, 0, 1, 3, 6, 6, 3, 1, 0, 1, 3, 9, 9, 9, 3, 1, 0, 1, 4, 12, 18, 18, 12, 4, 1, 0, 1, 4, 16, 24, 36, 24, 16, 4, 1, 0, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 0, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 0, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1

Offset: 0

Philippe Deléham, Jan 25 2010

Number of symmetric Dyck paths of semilength n with k peaks.

			Triangle begins :
  1;
  0,  1;
  0,  1,  1;
  0,  1,  1,  1;
  0,  1,  2,  2,  1;
  0,  1,  2,  4,  2,   1;
  0,  1,  3,  6,  6,   3,   1;
  0,  1,  3,  9,  9,   9,   3,   1;
  0,  1,  4, 12, 18,  18,  12,   4,   1;
  0,  1,  4, 16, 24,  36,  24,  16,   4,  1;
  0,  1,  5, 20, 40,  60,  60,  40,  20,  5,  1;
  0,  1,  5, 25, 50, 100, 100, 100,  50, 25,  5,  1;
  0,  1,  6, 30, 75, 150, 200, 200, 150, 75, 30,  6,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001405 (row sums), A005566, A084938, A088518 (diagonal sums), A088855.

Column k: A000007 (k=0), A000012 (k=1), A008619 (k=2), A002620 (k=3), A028724 (k=4), A028723 (k=5), A028725 (k=6), A331574 (k=7).

[n eq 0 select 1 else (&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_]:= Product[Binomial[Floor[(n-j)/2], Floor[(k-j)/2]], {j,0,1}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2022 *)

def A172101(n,k):
    if (n==0): return 1
    else: return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A172101(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 08 2022

Sum_{k=0..n} T(n,k) = A001405(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - [n=1] + A088518(n)*[n >= 1].
From G. C. Greubel, Apr 08 2022: (Start)
T(n, k) = binomial(floor((n-1)/2), floor((k-1)/2))*binomial(floor(n/2), floor(k/2)).
T(2*n, n) = [n=0] + A005566(n-1)*[n >= 1].
T(n-1, n-k) = T(n-1, k), n >= 1, 1 <= k <= n. (End)

A378069 a(n) = Sum_{k=0..n} binomial(n, floor(k/2 - 1/2)) * binomial(n, ceiling(k/2 - 1/2)).

Original entry on oeis.org

0, 1, 3, 13, 45, 181, 658, 2605, 9705, 38251, 144606, 569317, 2173262, 8556822, 32890068, 129565485, 500583105, 1973295775, 7654363750, 30194784763, 117497078842, 463820452602, 1809540528588, 7147843461733, 27946421773270, 110458073192026, 432648616232028

Offset: 0

Peter Luschny, Dec 14 2024

Cf. A007318, A005566, A378070.

a := n -> add(binomial(n, floor(k/2 - 1/2))*binomial(n, ceil(k/2 - 1/2)), k=0..n):
seq(a(n), n = 0..26);

A104856 Triangle read by rows: T(n,k) = binomial(n,k)binomial(k,floor(k/2))binomial(n-k,floor((n-k)/2)) (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 6, 6, 3, 6, 12, 24, 12, 6, 10, 30, 60, 60, 30, 10, 20, 60, 180, 180, 180, 60, 20, 35, 140, 420, 630, 630, 420, 140, 35, 70, 280, 1120, 1680, 2520, 1680, 1120, 280, 70, 126, 630, 2520, 5040, 7560, 7560, 5040, 2520, 630, 126, 252, 1260, 6300

Offset: 0

Emeric Deutsch, Apr 23 2005

T(n,k) is the number of paths in the first quadrant, starting from the origin, with unit steps up, down, right, or left, having a total of n steps, exactly k of which are vertical (up or down). Example: T(3,2)=6 because we have NNE, NEN, ENN, NSE, ENS and NES. [Emeric Deutsch, Nov 22 2008]

David M. Bloom et al., A Convolution of Middle Binomial Coefficients: Problem 10921, Amer. Math. Monthly 110, (2003), 958-959.
E. Deutsch and D. Lovit, Problem 1739, Math. Magazine, vol. 80, No. 1, 2007, p. 80. [_Emeric Deutsch_, Nov 22 2008]

Row sums yield A005566. T(n, 0)=T(n, n)=A001405(n).

Cf. A005566, A001405.

T:=(n,k)->binomial(n,k)*binomial(k,floor(k/2))*binomial(n-k,floor((n-k)/2)): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T(n, k) = binomial(n, k)*binomial(k, floor(k/2))*binomial(n-k, floor((n-k)/2)) (0<=k<=n).

A302185 Number of 3D n-step walks of type acc.

Original entry on oeis.org

1, 2, 7, 24, 98, 400, 1785, 7980, 37674, 178164, 874146, 4294752, 21667932, 109436184, 563910633, 2908233900, 15235550330, 79870553620, 424021948350, 2252356700880, 12088746573540, 64913104882080, 351594254659830, 1905139854213960, 10399223643879420, 56783986550235000

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A000918, A001405, A005558, A005566, A135394, A302182.

b:= n-> binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)):
C:= n-> binomial(2*n, n)/(n+1):
a:= n-> add(binomial(n, 2*k)*C(k)*b(n-2*k), k=0..n/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 24][n+1],
      (8*(14*n^4+85*n^3+190*n^2+188*n+63)*a(n-1)+4*(n-1)*
      (80*n^4+418*n^3+676*n^2+269*n-108)*a(n-2)-96*(n-1)*(n-2)*
      (10*n^2+31*n+27)*a(n-3)-144*(n-1)*(n-2)*(n-3)*(8*n^2+33*n+36)*
       a(n-4))/((n+4)*(n+3)*(n+2)*(8*n^2+17*n+11)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024

b[n_] := Binomial[n, Floor[n/2]]*Binomial[n+1, Floor[(n+1)/2]];
c[n_] := Binomial[2*n, n]/(n+1);
a[n_] := Sum[Binomial[n, 2*k]*c[k]*b[n - 2*k], {k, 0, n/2}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 28 2025, after Alois P. Heinz *)

from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def a(n):
    return sum(binomial(n, k)*C((k+1)//2)*C(k//2)*(2*(k//2)+1)*binomial(n-k, (n-k)//2) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Dec 06 2024

From Mélika Tebni, Dec 06 2024: (Start)
E.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*BesselI(1, 2*x) / x.
a(n) = Sum_{k=0..n} binomial(n, k)*A005558(k)*A001405(n-k).
a(2*n+1) = 2*A302182(2*n+1) = A135394(n) / (n+1).
For n > 0, a(A000918(n)) is odd. (End)

a(13)-a(25) from Mélika Tebni, Dec 06 2024

A381889 Expansion of e.g.f.: (BesselI(0, 2x) + BesselI(1, 2x))^2exp(2x).

Original entry on oeis.org

1, 4, 18, 86, 428, 2192, 11468, 60986, 328532, 1788368, 9819128, 54302712, 302157424, 1690193728, 9497996152, 53588976802, 303434431108, 1723578967056, 9818195961512, 56071829010968, 320970950634288, 1841213871449152, 10582333064327824, 60929582362628968, 351385363433883472

Offset: 0

Mélika Tebni, Mar 09 2025

nonn

Binomial transform of A151093.

For p prime, a(p) - 2 == 0 (mod 2*p).

Cf. A001405, A001700, A005566, A151093.

a := n-> add(binomial(n, k)*binomial(n-k, iquo(n-k,2))*binomial(2*k+1,k+1), k = 0 .. n): seq(a(n), n = 0 .. 24);

len := 24; Table[n!,{n, 0, len}] CoefficientList[Series[(BesselI[0, 2x] + BesselI[1, 2x])^2 Exp[2x], {x, 0, len}], x]  (* Peter Luschny, Mar 19 2025 *)

my(x='x+O('x^30)); Vec(serlaplace((besseli(0, 2*x) + x*besseli(1, 2*x))^2*exp(2*x))) \\ Michel Marcus, Mar 11 2025

from math import comb as C
def a(n):
    return sum(C(n, k)*2**(n-k)*C(k, k//2)*C(k+1, (k+1)//2) for k in range(n+1))
print([a(n) for n in range(25)])

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*A005566(k).
a(n) = Sum_{k=0..n} binomial(n, k)*A001405(n-k)*A001700(k).
a(n) = Sum_{k=0..n} binomial(n, k)*A005773(n-k+1)*A005773(k+1). - Mélika Tebni, Mar 19 2025

cp's OEIS Frontend

A064036 Number of walks of length n on cubic lattice, starting at origin, staying in first (nonnegative) octant.

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A360858 Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).

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A120406 Triangle read by rows: related to series expansion of the square root of 2 linear factors.

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Maxima

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A172101 Triangle, read by rows, given by [0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...] DELTA [1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, ...] where DELTA is the operator defined in A084938.

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A378069 a(n) = Sum_{k=0..n} binomial(n, floor(k/2 - 1/2)) * binomial(n, ceiling(k/2 - 1/2)).

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A104856 Triangle read by rows: T(n,k) = binomial(n,k)*binomial(k,floor(k/2))*binomial(n-k,floor((n-k)/2)) (0<=k<=n).

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A302185 Number of 3D n-step walks of type acc.

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A381889 Expansion of e.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*exp(2*x).

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A104856 Triangle read by rows: T(n,k) = binomial(n,k)binomial(k,floor(k/2))binomial(n-k,floor((n-k)/2)) (0<=k<=n).

A381889 Expansion of e.g.f.: (BesselI(0, 2x) + BesselI(1, 2x))^2exp(2x).