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

0,2

Comments

Hankel transform is A133460.

Links

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.

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Crossrefs

Row sums of A091811.

Sequence in context: A359713 A092636 A337928 * A007197 A247639 A002649

Adjacent sequences: A178789 A178790 A178791 * A178793 A178794 A178795

Keyword

nonn,easy

Author

Joseph Abate, Jun 15 2010

Status

approved