A069723 -id:A069723 - cp's OEIS frontend

A082142 Duplicate of A069723.

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368

Offset: 1

dead

A069720 a(n) = 2^(n-1)binomial(2n-1, n).

1, 6, 40, 280, 2016, 14784, 109824, 823680, 6223360, 47297536, 361181184, 2769055744, 21300428800, 164317593600, 1270722723840, 9848101109760, 76467608616960, 594748067020800, 4632774416793600, 36135640450990080, 282202144474398720, 2206307674981662720, 17266755717247795200

Offset: 1

Valery A. Liskovets, Apr 07 2002

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

First superdiagonal of number array A082137.

Cf. A000079, A001700, A003584, A069723, A069724, A082143.

a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1)
-- Reinhard Zumkeller, Jan 15 2015

[2^(n-2)*Binomial(2*n, n): n in [1..25]]; // Vincenzo Librandi, Apr 14 2018

Z:=(1-sqrt(1-2*z))*4^(n-1)/sqrt(1-2*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..20); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-1) Binomial[2n-1,n],{n,20}] (* Harvey P. Dale, Jan 20 2013 *)

a(n) = binomial(2*n-1,n)<<(n-1) \\ Charles R Greathouse IV, Feb 06 2017

def A069720(n): return 2^(n-2)*binomial(2*n, n)
print([A069720(n) for n in range(1,31)]) # G. C. Greubel, Jan 18 2025

a(n) = 2^(n-2)*binomial(2*n, n).
G.f.: (1-sqrt(1-8*x))/(4*x*sqrt(1-8*x)) = 2/(sqrt(1-8*x)*(1-sqrt(1-8*x))) - 1/(2*x). - Paul Barry, Sep 06 2004
D-finite with recurrence: n*a(n) - 4*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n! * [x^n] (exp(4*x)*BesselI(0, 4*x) - 1)/4. - Peter Luschny, Aug 25 2012
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A000079(n-1) * A001700(n-1); for n > 1:
a(n) = 2*A082143(n-1). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 4/7 + 32*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + 16*log(2)/27. (End)
a(n) = ((2*n)!/4) * [x^n] (BesselI(0, 2*sqrt(2*x)) - 1). - G. C. Greubel, Jan 18 2025

A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 8, 1, 5, 20, 40, 40, 16, 1, 6, 30, 80, 120, 96, 32, 1, 7, 42, 140, 280, 336, 224, 64, 1, 8, 56, 224, 560, 896, 896, 512, 128, 1, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 1, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512

Offset: 0

Paul Barry, Apr 06 2003

Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.
T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
Read as a triangle this is a subtriangle of A198793. - Philippe Deléham, Nov 10 2013

			Rows begin
  1 1  2   4   8 ...
  1 2  6  16  40 ...
  1 3 12  40 120 ...
  1 4 20  80 280 ...
  1 5 30 140 560 ...
Read as a triangle, this begins:
  1
  1, 1
  1, 2,  2
  1, 3,  6,  4
  1, 4, 12, 16,   8
  1, 5, 20, 40,  40, 16
  1, 6, 30, 80, 120, 96, 32
  ... - _Philippe Deléham_, Nov 10 2013

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A119468, A007318, A134309.

# As a triangular array:
T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, ] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* _Jean-François Alcover, Jan 20 2012 *)

def A082137_row(n) : # as a triangular array
    var('z')
    s = (exp(z*x)/(1-tanh(x))).series(x,n+2)
    t = factorial(n)*s.coefficient(x,n)
    return [t.coefficient(z,n-k) for k in (0..n)]
for n in (0..7) : print(A082137_row(n))  # Peter Luschny, Aug 01 2012

Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.
As an infinite lower triangular matrix, equals A007318 * A134309. - Gary W. Adamson, Oct 19 2007
O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - Peter Bala, Apr 26 2012
For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 10 2013

A119468 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(n,2j)*binomial(n-2j,k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 16, 12, 4, 1, 16, 40, 40, 20, 5, 1, 32, 96, 120, 80, 30, 6, 1, 64, 224, 336, 280, 140, 42, 7, 1, 128, 512, 896, 896, 560, 224, 56, 8, 1, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 1, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1

Offset: 0

Paul Barry, May 21 2006

Product of Pascal's triangle A007318 and A119467. Row sums are A007051. Diagonal sums are A113225.
Variant of A080928, A115068 and A082137. - R. J. Mathar, Feb 09 2010
Matrix inverse of the Euler tangent triangle A081733. - Peter Luschny, Jul 18 2012
Central column: T(2*n,n) = A069723(n). - Peter Luschny, Jul 22 2012
Subtriangle of the triangle in A198792. - Philippe Deléham, Nov 10 2013

			Triangle begins
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   16,   12,    4,    1;
   16,   40,   40,   20,    5,    1;
   32,   96,  120,   80,   30,    6,    1;
   64,  224,  336,  280,  140,   42,    7,   1;
  128,  512,  896,  896,  560,  224,   56,   8,  1;
  256, 1152, 2304, 2688, 2016, 1008,  336,  72,  9,  1;
  512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1;

A082137 read as triangle with rows reversed.

Cf. A000182, A009006, A038207, A081733, A155585.

A119468_row := proc(n) local s,t,k;
  s := series(exp(z*x)/(1-tanh(x)),x,n+2);
  t := factorial(n)*coeff(s,x,n); seq(coeff(t,z,k), k=(0..n)) end:
for n from 0 to 7 do A119468_row(n) od; # Peter Luschny, Aug 01 2012
# Alternatively:
T := (n, k) -> 2^(n-k-1+0^(n-k))*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

A[k_] := Table[If[m < n, 1, -1], {m, k}, {n, k}]; a = Join[{{1}}, Table[(-1)^n*CoefficientList[CharacteristicPolynomial[A[n], x], x], {n, 1, 10}]]; Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Jan 25 2009 *)
Table[Sum[Binomial[n,2j]Binomial[n-2j,k],{j,0,n-k}],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Dec 14 2022 *)

R = PolynomialRing(QQ, 'x')
def p(n,x) :
  return 1 if n==0 else add((-1)^n*binomial(n,k)*(x^(n-k)-1) for k in range(n))
def A119468_row(n):
    x = R.gen()
    return [abs(cf) for cf in list((p(n,x-1)-p(n,x+1))/2+x^n)]
for n in (0..8) : print(A119468_row(n)) # Peter Luschny, Jul 22 2012

G.f.: (1 - x - xy)/(1 - 2x - 2x*y + 2x^2*y + x^2*y^2).
Number triangle T(n,k) = Sum_{j=0..n} binomial(n,j)*binomial(j,k)*(1+(-1)^(j-k))/2.
Define matrix: A(n,m,k) = If[m < n, 1, -1];
  p(x,k) = CharacteristicPolynomial[A[n,m,k],x]; then t(n,m) = coefficients(p(x,n)). - Roger L. Bagula and Gary W. Adamson, Jan 25 2009
E.g.f.: exp(x*z)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2) for n >= 2, T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 10 2013
E.g.f.: [(e^(2t)+1)/2] e^(tx) = e^(P.(x)t), so this is an Appell sequence with lowering operator D = d/dx and raising operator R = x + 2/(e^(-2D)+1), i.e., D P_n(x) = n P_{n-1}(x) and R p_n(x) = P_{n+1}(x) where P_n(x) = [(x+2)^n + x^n]/2. Also, (P.(x)+y)^n = P_n(x+y), umbrally. R = x + 1 + D - 2 D^3/3! + ... contains the e.g.f.(D) mod signs of A009006 and A155585 and signed, aerated A000182, the zag numbers, so the unsigned differential component 2/[e^(2D)+1] = 2 Sum_{n >= 0} Eta(-n) (-2D)^n/n!, where Eta(s) is the Dirichlet eta function, and 2 *(-2)^n Eta(-n) = (-1)^n (2^(n+1)-4^(n+1)) Zeta(-n) = (2^(n+1)-4^(n+1)) B(n+1)/(n+1) with Zeta(s), the Riemann zeta function, and B(n), the Bernoulli numbers. The polynomials PI_n(x) of A081733 are the umbral compositional inverses of this sequence, i.e., P_n(PI.(x)) = x^n = PI_n(P.(x)) under umbral composition. Aside from the signs and the main diagonals, multiplying this triangle by 2 gives the face-vectors of the hypercubes A038207. - Tom Copeland, Sep 27 2015
T(n,k) = 2^(n-k-1+0^(n-k))*binomial(n, k). - Peter Luschny, Nov 10 2017

A082143 First subdiagonal of number array A082137.

Original entry on oeis.org

1, 3, 20, 140, 1008, 7392, 54912, 411840, 3111680, 23648768, 180590592, 1384527872, 10650214400, 82158796800, 635361361920, 4924050554880, 38233804308480, 297374033510400, 2316387208396800, 18067820225495040, 141101072237199360, 1103153837490831360

Offset: 0

Paul Barry, Apr 06 2003

			a(0)=(2^(-1)+(0^0)/2)C(1,0)=2*(1/2)=1 (use 0^0=1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A069723, A082144, A082145.

Cf. A000079, A001700, A069720.

a082143 0 = 1
a082143 n = (a000079 $ n - 1) * (a001700 n)
-- Reinhard Zumkeller, Jan 15 2015

[(2^(n-1) + 0^n/2)*Binomial(2*n+1,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Join[{1}, Table[2^(n-1)* Binomial[2*n+1,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+1,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(2n+1, n).
Conjecture: (n+1)*a(n) +4*(-2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 19 2014
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A000079(n-1) * A001700(n), for n > 0.
a(n) = A069720(n+1)/2. (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 64*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)) + 1/7.
Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/27 - 1/9. (End)

A098400 a(n) = 4^nbinomial(2n+1, n).

Original entry on oeis.org

1, 12, 160, 2240, 32256, 473088, 7028736, 105431040, 1593180160, 24216338432, 369849532416, 5671026163712, 87246556364800, 1346089726771200, 20819521107394560, 322702577164615680, 5011381198321090560, 77954818640550297600, 1214454016715941478400

Offset: 0

Paul Barry, Sep 06 2004

G. C. Greubel, Table of n, a(n) for n = 0..820

Cf. A000108, A001700, A069720, A069723, A098399, A151403.

[4^n*(2*n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Dec 27 2023

Table[4^n Binomial[2n+1,n],{n,0,20}] (* Harvey P. Dale, Jan 22 2019 *)

a(n)=binomial(2*n+1,n)<<(2*n) \\ Charles R Greathouse IV, Oct 23 2023

[4^n*binomial(2*n+1,n) for n in range(31)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-16*x))/(8*x*sqrt(1-16*x)).
E.g.f.: a(n) = n! * [x^n] exp(8*x)*(BesselI(0, 8*x) + BesselI(1, 8*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 8*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 4^n*(2*n+1)*Hypergeometric2F1([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 4^n * A001700(n).
a(n) = 4^n * (2*n+1) * A000108(n).
a(n) = (2*n+1) * A151403(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 8/15 + 128*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 8/17 + 128*arcsinh(1/4)/(17*sqrt(17)). (End)

A069722 Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400, 144542561803960320, 1128808577897594880

Offset: 1

Valery A. Liskovets, Apr 07 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A059304, A069720, A069721, A069723, A089156.

[0] cat[2^(n-1)*Binomial(2*n-2, n-1): n in [2..20]]; // Vincenzo Librandi, Nov 17 2011

Z:=(1-sqrt(1-z))*8^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..19); # Zerinvary Lajos, Jan 01 2007

Join[{0},Table[2^(n-1) Binomial[2n-2,n-1],{n,2,20}]] (* Harvey P. Dale, Nov 16 2011 *)

a(n) = 2^(n-1)*binomial(2n-2, n-1), n>1.
a(n) = 2*A069723(n), n>1.
G.f. for a(n)^2: 1/AGM(1, (1-64*x)^(1/2)). - Benoit Cloitre, Jan 01 2004
a(n) = A059304(n-1), n>1. [R. J. Mathar, Sep 29 2008]
a(n) ~ 2^(3*n-3)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 28 2019
E.g.f.: x * (exp(4*x) * (BesselI(0,4*x) - BesselI(1,4*x)) - 1). - Ilya Gutkovskiy, Nov 03 2021
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=2} 1/a(n) = 1/7 + 8*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=2} (-1)^n/a(n) = 1/9 + 4*log(2)/27. (End)

A082144 A subdiagonal of number array A082137.

Original entry on oeis.org

1, 4, 30, 224, 1680, 12672, 96096, 732160, 5601024, 42997760, 331082752, 2556051456, 19778969600, 153363087360, 1191302553600, 9268801044480, 72219408138240, 563445537177600, 4401135695953920, 34414895667609600, 269374774271016960, 2110381254330286080

Offset: 0

Paul Barry, Apr 06 2003

			a(0)=(2^(-1)+(0^0)/2)C(2,0)=2*(1/2)=1 (use 0^0=1).

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

Cf. A069723, A082137, A082143, A082145.

[(2^(n-1) + 0^n/2)*Binomial(2*n+2,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Join[{1}, Table[2^(n-1)*Binomial[2*n+2, n], {n,1,50}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+2,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(2*n+2, n).
(n+2)*a(n) +12*(-n-1)*a(n-1) +16*(2*n-1)*a(n-2)=0. - R. J. Mathar, Oct 29 2014
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 88*arccot(sqrt(7))/(7*sqrt(7)) - 3/7.
Sum_{n>=0} (-1)^n/a(n) = 52*log(2)/27 - 5/9. (End)

A082145 A subdiagonal of number array A082137.

Original entry on oeis.org

1, 5, 42, 336, 2640, 20592, 160160, 1244672, 9674496, 75246080, 585761792, 4564377600, 35602145280, 277970595840, 2172375244800, 16992801914880, 133035751833600, 1042374243778560, 8173537721057280, 64136851016908800, 503613708419727360, 3956964851869286400

Offset: 0

Paul Barry, Apr 06 2003

			a(0) = ( 2^(-1)+(0^0)/2 )*C(3,0) = ( 1/2+1/2 )*1 = 1 (use 0^0 = 1). - clarified by _Jon Perry_, Oct 29 2014

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

Cf. A069723, A082143, A082144.

[(2^(n-1)+(0^n)/2)*Binomial(2*n+3, n): n in [0..30]]; // Vincenzo Librandi, Oct 30 2014

Z:=(1-3*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=4..23); # Zerinvary Lajos, Jan 01 2007

Join[{1}, Table[2^(n-1)* Binomial[2*n+3,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+3,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = ( 2^(n-1) + (0^n)/2 )*binomial(2*n+3, n).
(n+3)*a(n) +2*(-7*n-13)*a(n-1) +24*(2*n+1)*a(n-2)=0. - R. J. Mathar, Oct 29 2014
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 37/7 - 208*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 296*log(2)/27 - 61/9. (End)

A098399 a(n) = 3^nbinomial(2n+1, n).

Original entry on oeis.org

1, 9, 90, 945, 10206, 112266, 1250964, 14073345, 159497910, 1818276174, 20827527084, 239516561466, 2763652632300, 31979409030900, 370961144758440, 4312423307816865, 50227047938102310, 585982225944526950, 6846739692614999100, 80106854403595489470, 938394580156404305220

Offset: 0

Paul Barry, Sep 06 2004

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

Cf. A000108, A001700, A005159, A069720, A069723, A098400.

[3^n*Binomial(2*n+1, n): n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Z:=(1-sqrt(1-3*z))*4^n/sqrt(1-3*z)/6: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..18); # Zerinvary Lajos, Jan 01 2007

Table[3^n Binomial[2n+1,n], {n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)

a(n)=binomial(2*n+1,n)*3^n \\ Charles R Greathouse IV, Oct 23 2023

[3^n*binomial(2*n+1, n) for n in range(21)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-12*x))/(6*x*sqrt(1-12*x)).
E.g.f.: a(n) = n!* [x^n] exp(6*x)*(BesselI(0, 6*x) + BesselI(1, 6*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 6*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 3^n * (2*n+1)*A000108(n).
a(n) = (2*n+1)*A005159(n).
a(n) = 3^n * A001700(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 6/11 + 72*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 6/13 + 72*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)

cp's OEIS Frontend

A082142 Duplicate of A069723.

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A069720 a(n) = 2^(n-1)*binomial(2*n-1, n).

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A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

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A119468 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(n,2j)*binomial(n-2j,k).

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A082143 First subdiagonal of number array A082137.

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A098400 a(n) = 4^n*binomial(2*n+1, n).

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A069722 Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A069720 a(n) = 2^(n-1)binomial(2n-1, n).

A098400 a(n) = 4^nbinomial(2n+1, n).

A098399 a(n) = 3^nbinomial(2n+1, n).