A069720 -id:A069720 - 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)

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)

A098660 E.g.f. BesselI(0,2sqrt(2)x) + BesselI(1,2sqrt(2)x)/sqrt(2).

Original entry on oeis.org

1, 1, 4, 6, 24, 40, 160, 280, 1120, 2016, 8064, 14784, 59136, 109824, 439296, 823680, 3294720, 6223360, 24893440, 47297536, 189190144, 361181184, 1444724736, 2769055744, 11076222976, 21300428800, 85201715200, 164317593600

Offset: 0

Paul Barry, Sep 20 2004

Third binomial transform (shifted right) is A047781. Hankel transform is A166232(n+1).

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

Cf. A059304, A069720 (bisections).

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+4*x-Sqrt(1-8*x^2))/(4*x*Sqrt(1-8*x^2)))); // G. C. Greubel, Aug 17 2018

nmax = 30; CoefficientList[Series[BesselI[0, 2*Sqrt[2]*x] + BesselI[1, 2*Sqrt[2]*x]/Sqrt[2], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 13 2017 *)

x='x+O('x^30); Vec((1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2))) \\ G. C. Greubel, Aug 17 2018

G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)) = (1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)).
a(n) = binomial(n, floor(n/2))2^floor(n/2).
a(n+1) = (1/Pi)*int(x^n*(x+4)/sqrt(8-x^2),x,-2*sqrt(2),2*sqrt(2)) if n is odd [corrected by Vaclav Kotesovec, Nov 13 2017].
Conjecture: (n+1)*a(n) +(n-1)*a(n-1) -n*a(n-2) +(2-n)*a(n-3) = 0. - R. J. Mathar, Nov 15 2011

A135838 Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 12, 12, 4, 4, 16, 24, 16, 4, 8, 40, 80, 80, 40, 8, 8, 48, 120, 160, 120, 48, 8, 16, 112, 336, 560, 560, 336, 112, 16, 16, 128, 448, 896, 1120, 896, 448, 128, 16, 32, 288, 1152, 2688, 4032, 4032, 2688, 1152, 288, 32

Offset: 1

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle are:
  1;
  2,  2;
  2,  4,  2;
  4, 12, 12,  4;
  4, 16, 24, 16,  4;
  8, 40, 80, 80, 40, 8;
  ...

Gheorghe Coserea, Rows n = 1..100, flattened

Cf. A016116, A059304, A069720, A093968, A094015 (row sums), A135837.

A135838 := proc(n,k)
    2^floor(n/2)*binomial(n-1,k-1) ;
end proc:
seq(seq( A135838(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 15 2022

T[n_, k_]:= 2^Floor[n/2]*Binomial[n-1, k-1];
Table[T[n, k], {n,12}, {k,n}] //Flatten (* G. C. Greubel, Feb 07 2022 *)

A(n,k) = 2^(n\2)*binomial(n-1,k-1);
concat(vector(10, n, vector(n, k, A(n,k))))  \\ Gheorghe Coserea, May 18 2016

flatten([[2^(n//2)*binomial(n-1, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 07 2022

M * Pascal's triangle as infinite lower triangular matrices, where M = a triangle with (1, 2, 2, 4, 4, 8, 8, 16, 16, ...) in the main diagonal and the rest zeros.
Sum_{k=1..n} T(n, k) = A094015(n-1).
From G. C. Greubel, Feb 07 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A016116(n).
T(n, 2) = 2*A093968(n-1).
T(2*n-1, n) = A059304(n-1).
T(2*n, n) = 2*A069720(n). (End)

A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 5, 28, 168, 1056, 6864, 45760, 311168, 2149888, 15049216, 106502144, 760729600, 5477253120, 39710085120, 289650032640, 2124100239360, 15651264921600, 115819360419840, 860372391690240

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A069720, A000108, A003645.

Cf. A000346, A001006, A136576.

Z:=-(1-4*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=3..24); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = 2^(n-2)*C_(n+1), where C_n stands for the Catalan numbers (A000108).
a(n) = A003645(n+2)/4.
D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021
From Peter Bala, Apr 29 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1).
O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).
A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)

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

A103944 Number of rooted unicursal n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 10, 93, 836, 7355, 63750, 546553, 4646920, 39250935, 329789450, 2758868981, 22995369996, 191074697203, 1583463268366, 13092015636465, 108024564809744, 889730213085167, 7316434446188562, 60078376613838829, 492692533579612180

Offset: 1

Valery A. Liskovets, Mar 17 2005

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A069720, A103945.

Flatten[{1,Table[n*Binomial[2n,n]*Sum[Binomial[n-2,k]*(1/(n+1+k)+n/(n+2+k)),{k,0,n-2}],{n,2,20}]}] (* Vaclav Kotesovec, Oct 17 2012 *)

a(n)=n*binomial(2n, n)sum_{i=0..n-2} binomial(n-2, i)(1/(n+1+i)+n/(n+2+i)), for n>1.
Recurrence: (n-1)*a(n) = 3*(3*n-4)*a(n-1) - 6*(n-9)*a(n-2) - 8*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 8^n*sqrt(n)/(6*sqrt(Pi)). - Vaclav Kotesovec, Oct 17 2012

A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Philippe Deléham, Dec 14 2011

A071919*A007318 as infinite lower triangular matrices.
A129186*A038207 as infinite lower triangular matrices.
From Paul Curtz, Nov 12 2019: (Start)
If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
   The first column is A166444.
   The second column is A139756.
   The antidiagonal sums are A000129 (Pell numbers).
   The row sums are (-1)^n*A141413.
   The signed row sums are 0 followed by 1's, autosequence companion to A054977.
(End)

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...

Essentially the same as A038207, A062715, A065109.
Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
Cf. A141413, A166444, A054977.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
G.f.: (1-(1+y)*x)/(1-(2+y)*x).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
T(2n,n) = A069720(n).

A003584 Unicursal (i.e., possessing an Eulerian path) planar rooted maps with n edges.

Original entry on oeis.org

1, 2, 9, 52, 336, 2304, 16368, 118976, 878592, 6562816, 49447424, 375072768, 2860343296, 21909012480, 168425533440, 1298753372160, 10041201131520, 77809145610240, 604138825973760, 4698956908462080, 36604934482821120

Offset: 0

Valery A. Liskovets

Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is maps possessing an Eulerian path or cycle).

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000257, A005470, A069720.

a[n_] := 2^(n-1)*(3*Binomial[2*n, n]/((n+1)*(n+2))+Binomial[2*n-1, n]); a[0]=1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2014 *)

a(n) = A000257(n) + A069720(n).

More terms from Valery A. Liskovets, Apr 07 2002

A098405 Expansion of (1-sqrt(1-8x))/((1-x)(4xsqrt(1-8*x))).

Original entry on oeis.org

1, 7, 47, 327, 2343, 17127, 126951, 950631, 7173991, 54471527, 415652711, 3184708455, 24485137255, 188802730855, 1459525454695, 11307626564455, 87775235181415, 682523302202215, 5315297718995815, 41450938169985895, 323653082644384615, 2529960757626047335

Offset: 0

Paul Barry, Sep 06 2004

Partial sums of A069720.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A069720.

[n le 2 select 7^(n-1) else ((9*n-4)*Self(n-1) - 4*(2*n-1)*Self(n-2))/n: n in [1..30]]; // G. C. Greubel, Dec 27 2023

Table[SeriesCoefficient[(1-Sqrt[1-8*x])/((1-x)*(4*x*Sqrt[1-8*x])),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
Accumulate[Table[2^(n-1) Binomial[2n-1,n],{n,20}]] (* Harvey P. Dale, Jan 20 2013 *)

my(x='x+O('x^66)); Vec((1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x)))) \\ Joerg Arndt, May 11 2013

def A098405(n): return sum( binomial(2*k+1,k)*2^k for k in range(n+1))
[A098405(n) for n in range(41)] # G. C. Greubel, Dec 27 2023

a(n) = Sum_{k=0..n} binomial(2*k+1, k)*2^k.
D-finite with recurrence: (n+1)*a(n) = (9*n+5)*a(n-1) - 4*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ 2^(3*n+4)/(7*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

cp's OEIS Frontend

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

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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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A098660 E.g.f. BesselI(0,2*sqrt(2)*x) + BesselI(1,2*sqrt(2)*x)/sqrt(2).

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A135838 Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).

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A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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

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A103944 Number of rooted unicursal n-edge maps in the plane (planar with a distinguished outside face).

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A098660 E.g.f. BesselI(0,2sqrt(2)x) + BesselI(1,2sqrt(2)x)/sqrt(2).

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

A098405 Expansion of (1-sqrt(1-8x))/((1-x)(4xsqrt(1-8*x))).