A133460 -id:A133460 - cp's OEIS frontend

A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

1, 5, 31, 209, 1471, 10625, 78079, 580865, 4361215, 32978945, 250806271, 1916280833, 14698053631, 113104519169, 872801042431, 6751535300609, 52337071357951, 406468580343809, 3162019821780991, 24634626678980609, 192179216026959871

Offset: 0

Joseph Abate, Jun 15 2010

Hankel transform is A133460.

			a(3) = (1,4,10,20)dot(1,2,4,8) = 209.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian motion and generalized Catalan numbers, Journal of Integer Sequences, Vol. 14 (2011).
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.

Row sums of A091811.

Cf. A383326, A383716.

a := n -> binomial(2*n+2,n+1)*hypergeom([-n, n + 1], [n + 2], -1)/2:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 21 2017

CoefficientList[Series[(3-Sqrt[1-8*x])/(2*(1+x)*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
Table[Sum[2^k*Binomial[n + k, k], {k, 0, n}], {n, 0, 20}] (* Michael De Vlieger, Oct 28 2016 *)
a[n_] := (-1)^(n + 1) - 2^(n + 1) (2n + 1) Binomial[2n, n] Hypergeometric2F1[1, 2n + 2, n + 2, 2]/(n + 1); Array[a, 22, 0] (* Robert G. Wilson v, Jul 21 2018 *)

a(n) = Sum_{k = 0..n} A046899(n,k)*2^k = Sum_{k = 0..n} 2^k * binomial(n+k,k).
G.f.: (1/3)*(4/sqrt(1 - 8*x) - 1/(1 - x*c(2*x))) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = (1/3)*(4*2^n*A000984(n) - A064062(n)).
a(n) + a(n+1) = 6*2^n*A001700(n).
O.g.f.: (3 - sqrt(1 - 8*x))/(2*(1 + x)*sqrt(1 - 8*x)). - Peter Bala, Apr 10 2012
a(n) = 2^n *binomial(2+2*n,1+n)*2F1(1, 2+2*n; 2+n;-1). - Olivier Gérard, Aug 19 2012
D-finite with recurrence n*a(n) = (7*n - 4)*a(n-1) + 4*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(3n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k = 0..n} binomial(k+n,n) * binomial(2*n+1,n-k). - Vladimir Kruchinin, Oct 28 2016
a(n) = 1/2*(n + 1)*binomial(2*n+2,n+1)*Sum_{k = 0..n} binomial(n,k)/(n + k + 1). - Peter Bala, Feb 21 2017
a(n) = binomial(2*n+2,n+1)*hypergeom([-n, n+1], [n+2], -1)/2. - Peter Luschny, Feb 21 2017
a(n) = (-1)^(n+1) - 2^(n+1)*(2*n+1)*binomial(2*n,n)*hypergeom([1, 2*n+2], [n+2], 2)/(n+1). - John M. Campbell, Jul 14 2018
From Akiva Weinberger, Dec 06 2024: (Start)
a(n) = (2*n + 1)!/(n!^2) * Integral_{t=0..1} (t + t^2)^n dt.
a(n) = (Integral_{t=0..1} (t + t^2)^n dt) / (Integral_{t=0..1} (t - t^2)^n dt). (End)
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*n+1,k) * (-2)^k. - André M. Timpanaro, Dec 15 2024
From Seiichi Manyama, Aug 04 2025: (Start)
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(n+1)). (End)

A089022 Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex.

Original entry on oeis.org

1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999, 8099511, 57959535, 418441191, 3043608351, 22280372247, 164008329423, 1213166815047, 9012417249663, 67208553680247, 502920171632943, 3775020828459687, 28415858155984863, 214444848602732247, 1622146752543427983

Offset: 0

Paul Boddington, Nov 11 2003

The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients.

Hankel transform is A133460. - Philippe Deléham, Dec 01 2007

			a(2) = 15 because there are 3*3=9 walks whose second step is to return to the starting vertex and 3*2=6 walks whose second step is to move away from the starting vertex.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.
Ed Pegg Jr., K-Cayley Trees

Column k=3 of A183135.

A000602 := x -> 2^x*binomial(2*x, x)-9^x+1/3*2^x*binomial(2*x, x) * hypergeom([1, 2*x+1], [x+1], 2/3); # Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007

Table[2^n*Binomial[2*n,n]-3^(n-1)*Sum[(2/3)^k*Binomial[n+k,n],{k,0,n-1}],{n,0,20}] (* or *)
CoefficientList[Series[4/(1+3*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

my(x='x+O('x^30)); Vec(4/(1+3*sqrt(1-8*x))) \\ Joerg Arndt, May 10 2013

G.f.: 4/(1+3*sqrt(1-8*x)).
a(n) = 2^n * binomial(2*n,n) - 3^(n-1) * Sum_{j=0..n-1} (2/3)^j*binomial(n+j,n). - Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007
a(n) = Sum_{k=0..n} A039599(n,k)*2^(n-k). - Philippe Deléham, Aug 25 2007
From Paul Barry, Sep 04 2009: (Start)
G.f.: 1/(1-3x/(1-2x/(1-2x/(1-2x/(1-2x/(1-... (continued fraction);
G.f.: (1-2*x*c(x))/(1-3*x-2*x*c(x)), where c(x) is the g.f. of A000108. (End)
a(n) = A126087(2n). - Philippe Deléham, Nov 02 2011
D-finite with recurrence n*a(n) + (12-17*n)*a(n-1) + 36*(2n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 6*8^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
From Karol A. Penson, Sep 06 2014: (Start)
a(n) is the (2*n)-th moment of a positive function W(x) = (3/Pi)*sqrt(8-x^2)/(9-x^2), on the segment x = (0,2*sqrt(2)): a(n) = Integral_{x=0..2*sqrt(2)} x^(2*n)*W(x) dx;
a(n) is the special value of hypergeometric function 2F1, in Maple notation: a(n)=2*8^n*GAMMA(n+1/2)*hypergeom([1,n+1/2],[n+2],8/9)/(3*sqrt(Pi)*(n+1)!). (End)
a(n) = A151374(n)*hypergeom([1,n+1/2],[n+2],8/9)*(2/3). - Peter Luschny, Sep 06 2014

cp's OEIS Frontend

A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

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