A069723 -id:A069723 - cp's OEIS frontend

A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

1, 4, 48, 640, 8960, 129024, 1892352, 28114944, 421724160, 6372720640, 96865353728, 1479398129664, 22684104654848, 348986225459200, 5384358907084800, 83278084429578240, 1290810308658462720, 20045524793284362240, 311819274562201190400, 4857816066863765913600

Offset: 0

Paul Barry, Sep 06 2004

It seems that a(n) is the number of pairs of binary vectors of length 2*n which are orthogonal. (Define binary vectors here to have components of value +1 or -1. There are no pairs of binary vectors of odd length which are orthogonal.) For example, the a(1) = 4 pairs of binary vectors of length 2 are (-1,-1) and (1,-1), (-1,-1) and (-1,1), (1,-1) and (1,1), (-1,1) and (1,1). Tested up to and including a(8). - R. J. Mathar, Apr 15 2013

Tested up to and including a(210). - R. H. Hardin, May 08 2013

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A055372, A069723, A069720, A098400, A098401.

[(0^n + 4^n*(n+1)*Catalan(n))/2: n in [0..40]]; // G. C. Greubel, Dec 27 2023

Table[(Boole[n == 0] + 4^n Binomial[2 n, n])/2, {n, 0, 18}] (* or *)
CoefficientList[Series[8 x/(# (1 - #)) &@ Sqrt[1 - 16 x], {x, 0, 18}], x] (* Michael De Vlieger, Aug 03 2016 *)

[(4^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

G.f.: 8*x/( sqrt(1 - 16*x)*(1 - sqrt(1 - 16*x)) ).
a(n+1) = 4*A098400(n).
n*a(n) - 8*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ 16^n/(2*sqrt(Pi*n)). - Ilya Gutkovskiy, Aug 03 2016
a(n) = A055372(2*n,n). - Alois P. Heinz, Jan 21 2020
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 17/15 + 32*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 15/17 - 32*arcsinh(1/4)/(17*sqrt(17)). (End)

A099045 a(n) = (30^n + 4^nbinomial(2*n,n))/4.

Original entry on oeis.org

1, 2, 24, 320, 4480, 64512, 946176, 14057472, 210862080, 3186360320, 48432676864, 739699064832, 11342052327424, 174493112729600, 2692179453542400, 41639042214789120, 645405154329231360, 10022762396642181120, 155909637281100595200

Offset: 0

Paul Barry, Sep 24 2004

(1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A069723, A088218, A099044, A099046.

[(3*0^n + 4^n*Binomial(2*n, n))/4: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Join[{1}, Table[4^(n-1)*Binomial[2*n,n], {n,1,30}]] (* G. C. Greubel, Dec 31 2017 *)

for(n=0,30, print1((3*0^n + 4^n*binomial(2*n,n))/4, ", ")) \\ G. C. Greubel, Dec 31 2017

G.f.: (1+3*sqrt(1-16*x))/(4*sqrt(1-16*x)).
n*a(n) +8*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012
E.g.f.: (3 + exp(8*x) * BesselI(0,8*x)) / 4. - Ilya Gutkovskiy, Nov 17 2021

A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 0, 4, 40, 336, 2688, 21120, 164736, 1281280, 9957376, 77395968, 601968640, 4686094336, 36515020800, 284817162240, 2223764766720, 17379001958400, 135942415319040, 1064286014668800, 8338993950228480, 65388301768458240, 513094808135270400, 4028909667357818880

Offset: 1

Valery A. Liskovets, Apr 07 2002

			G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...

Robert Israel, Table of n, a(n) for n = 1..1109
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics 339.11 (2016): 2652-2659.

Cf. A069720, A069722, A069723.

[0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // Vincenzo Librandi, Nov 13 2016

0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # Robert Israel, Nov 12 2016

a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

a(n) = 2^(n-2)*(n-2)*binomial(2n-2, n-1)/n, n>1.
From Robert Israel, Nov 12 2016: (Start)
G.f.: 32*x^3/(sqrt(1-8*x)*(1+sqrt(1-8*x))^3).
E.g.f.: ((1-6*x)/4)*exp(4*x)*I_0(4*x)+(3/2)*exp(4*x)*I_1(4*x)+x/2-1/4, where I_0 and I_1 are modified Bessel functions of the first kind.
a(n+1) = (4*(n-1)*(2*n-1)/((n+1)*(n-2)))*a(n).
a(n) ~ 8^n/(16*sqrt(Pi*n)). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=3} 1/a(n) = 11/14 - 26*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 37*log(2)/27 - 13/18. (End)

A099044 a(n) = (20^n + 3^nbinomial(2*n,n))/3.

Original entry on oeis.org

1, 2, 18, 180, 1890, 20412, 224532, 2501928, 28146690, 318995820, 3636552348, 41655054168, 479033122932, 5527305264600, 63958818061800, 741922289516880, 8624846615633730, 100454095876204620, 1171964451889053900, 13693479385229998200, 160213708807190978940

Offset: 0

Paul Barry, Sep 24 2004

(1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A069723, A088218, A099045, A099046.

[(2*0^n + 3^n*Binomial(2*n, n))/3: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Join[{1}, Table[3^(n-1)*binomial(2*n,n), {n,1,30}]] (* G. C. Greubel, Dec 31 2017 *)

for(n=0, 30, print1((2*0^n + 3^n*binomial(2*n,n))/3, ", ")) \\ G. C. Greubel, Dec 31 2017

G.f.: 1/3 + 4*x/(sqrt(1-12*x)(1-sqrt(1-12*x)))  = (1 + 2*sqrt(1-12*x))/(3*sqrt(1-12*x)).
n*a(n) +6*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012
E.g.f.: (2 + exp(6*x) * BesselI(0,6*x)) / 3. - Ilya Gutkovskiy, Nov 17 2021

A099046 a(n) = (40^n + 5^nbinomial(2*n,n))/5.

Original entry on oeis.org

1, 2, 30, 500, 8750, 157500, 2887500, 53625000, 1005468750, 18992187500, 360851562500, 6888984375000, 132038867187500, 2539208984375000, 48970458984375000, 946762207031250000, 18343517761230468750, 356080050659179687500, 6923778762817382812500

Offset: 0

Paul Barry, Sep 24 2004

(1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

G. C. Greubel, Table of n, a(n) for n = 0..760

Cf. A069723, A088218, A099044, A099045.

[(4*0^n + 5^n*Binomial(2*n, n))/5: n in [ 0..30]]; // G. C. Greubel, Dec 31 2017

CoefficientList[Series[(1+4Sqrt[1-20x])/(5Sqrt[1-20x]),{x,0,20}],x]  (* Harvey P. Dale, Mar 30 2011 *)
Join[{1}, Table[5^(n - 1)*Binomial[2*n, n], {n,1,50}]] (* G. C. Greubel, Dec 31 2017 *)

for(n=0,30, print1((4*0^n + 5^n*binomial(2*n,n))/5, ", ")) \\ G. C. Greubel, Dec 31 2017

G.f.: (1 + 4*sqrt(1-20*x))/(5*sqrt(1-20*x)).
n*a(n) +10*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012
E.g.f.: (4 + exp(10*x) * BesselI(0,10*x)) / 5. - Ilya Gutkovskiy, Nov 17 2021
a(n) = Integral_{x = 0..20} x^n * w(x) dx for n >= 1, where w(x) = 1/( 5*Pi*sqrt(x*(20 - x)) ) is positive on the interval (0, 20). The weight function w(x) is singular at x = 0 and at x = 20 and is the solution of the Hausdorff moment problem. - Peter Bala, Oct 12 2024

A128413 Inverse of number triangle A128412.

Original entry on oeis.org

1, 2, 1, 12, 8, 1, 80, 60, 12, 1, 560, 448, 112, 16, 1, 4032, 3360, 960, 180, 20, 1, 29568, 25344, 7920, 1760, 264, 24, 1, 219648, 192192, 64064, 16016, 2912, 364, 28, 1, 1647360, 1464320, 512512, 139776, 29120, 4480, 480, 32, 1, 12446720

Offset: 0

Paul Barry, Mar 02 2007

First column is A069723.

			Triangle begins
1,
2, 1,
12, 8, 1,
80, 60, 12, 1,
560, 448, 112, 16, 1,
4032, 3360, 960, 180, 20, 1,
29568, 25344, 7920, 1760, 264, 24, 1,
219648, 192192, 64064, 16016, 2912, 364, 28, 1

Cf. A128417.

Number triangle T(n,k)=if(k=0,2^n*(C(2n,n)/2+0^n/2),2^(n-k)*C(2n,n-k)); Column k has g.f. if(k=0,4x/(sqrt(1-8x)(1-sqrt(1-8x))),(1/sqrt(1-8x))*((1-4x-sqrt(1-8x))/(8x))^k);

A370280 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.

Original entry on oeis.org

1, 3, 25, 234, 2305, 23373, 241486, 2527920, 26720529, 284555700, 3048323135, 32812937820, 354619072990, 3845377105794, 41817926091120, 455893204069944, 4980851709418353, 54521955043418925, 597823622561048020, 6564929893462467450, 72189820135528858455

Offset: 0

Seiichi Manyama, Feb 13 2024

nonn

G. C. Greubel, Table of n, a(n) for n = 0..940

Cf. A069723, A249924, A370282.

R:=PowerSeriesRing(Rationals(), 100);
A370280:= func< n | Coefficient(R!( 1/(1-3*x+x^2)^n ), n) >;
[A370280(n): n in [0..30]]; // G. C. Greubel, Feb 07 2025

A370280[n_]:= Coefficient[Series[1/(1-3*x+x^2)^n, {x,0,100}], x, n];
Table[A370280[n], {n,0,40}] (* G. C. Greubel, Feb 07 2025 *)

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(3*n+k-1, n-k));

def A370280(n): return sum(binomial(n+j-1,j)*binomial(3*n+j-1,n-j) for j in range(n+1))
print([A370280(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(3*n+k-1,n-k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * ((1-x)^2 - x) ).
a(n) ~ sqrt((4 + sqrt(6))/(24*Pi*n)) * ((27 + 12*sqrt(6))/5)^n. - Vaclav Kotesovec, Feb 07 2025

A098401 a(n) = (0^n + 3^nbinomial(2n,n))/2.

Original entry on oeis.org

1, 3, 27, 270, 2835, 30618, 336798, 3752892, 42220035, 478493730, 5454828522, 62482581252, 718549684398, 8290957896900, 95938227092700, 1112883434275320, 12937269923450595, 150681143814306930, 1757946677833580850, 20540219077844997300, 240320563210786468410

Offset: 0

Paul Barry, Sep 06 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A069723, A069720, A098399, A098400, A098402.

[(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* Harvey P. Dale, Nov 29 2023 *)
Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* G. C. Greubel, Dec 27 2023 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n + 1}, {1}, 1]; Flatten[Table[a[n], {n,0,20}]] (* Detlef Meya, May 21 2024 *)

[(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

a(n+1) = 3*A098399(n).
G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))).
n*a(n) - 6*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)
a(n) = 3^n*hypergeom([-n, -n + 1], [1], 1). - Detlef Meya, May 21 2024

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

Original entry on oeis.org

1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600

Offset: 0

Peter Luschny, Apr 14 2014

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

Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.

A240558 := n -> 2^n*n!/((iquo(n,2)+1)*iquo(n,2)!^2):
seq(A240558(n), n=0..30);

Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x,0,n}], {n,0,22}]

x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017

def A240558():
    x, n = 1, 1
    while True:
        yield x
        m = 2*n if is_odd(n) else 8/(n+2)
        x *= m
        n += 1
a = A240558(); [next(a) for i in range(36)]

O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
For a recurrence see the Sage program.
a(n) = 2^n*A057977(n)
a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).
a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).
From Peter Luschny, Jan 31 2015: (Start)
a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
a(n) ~ 2^(n+N)/((n+1)^*sqrt(Pi*(2*N+1))); here  = 1 if n is even, 0 otherwise and N = n++1. (End)
Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

A370282 Coefficient of x^n in the expansion of 1/( (1-x)^3 - x )^n.

Original entry on oeis.org

1, 4, 42, 499, 6250, 80634, 1060269, 14127852, 190102482, 2577310285, 35150819132, 481734467955, 6628611532621, 91517611501008, 1267182734325900, 17589579427715124, 244689432718144770, 3410399867585709501, 47613678409439712861, 665756829352248572725

Offset: 0

Seiichi Manyama, Feb 13 2024

nonn

Cf. A069723, A370280.

Cf. A369215.

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(4*n+2*k-1, n-k));

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(4*n+2*k-1,n-k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * ((1-x)^3 - x) ). See A369215.

cp's OEIS Frontend

A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

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A099045 a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.

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A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A099044 a(n) = (2*0^n + 3^n*binomial(2*n,n))/3.

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A099046 a(n) = (4*0^n + 5^n*binomial(2*n,n))/5.

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A128413 Inverse of number triangle A128412.

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A370280 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.

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A099045 a(n) = (30^n + 4^nbinomial(2*n,n))/4.

A099044 a(n) = (20^n + 3^nbinomial(2*n,n))/3.

A099046 a(n) = (40^n + 5^nbinomial(2*n,n))/5.

A098401 a(n) = (0^n + 3^nbinomial(2n,n))/2.

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).