A102071 Pairwise sums of general ballot numbers (A002026).

1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496, 35142703099692, 101055137177563

Offset: 1

Author

Ralf Stephan, Dec 30 2004

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020. See Table 2.

Crossrefs

First differences of A005554. Partial sums of A026269. 3rd column of A348840.

Cf. A000108, A001006, A002057.

Programs

Mathematica

CoefficientList[Series[(4x(1+x))/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,40}],x] (* Harvey P. Dale, Feb 26 2013 *)

Maxima

a(n):=1/n*sum((binomial(j,n-1-j)+4*binomial(j,n-2-j)+3*binomial(j,n-3-j))*binomial(n,j),j,0,n); /* Vladimir Kruchinin, Mar 08 2016 */

PARI

z='z+O('z^66); Vec(4*z*(1+z)/(1-z+sqrt(1-2*z-3*z^2))^2) \\ Joerg Arndt, Mar 08 2016

Formula

G.f.: (4*x*(1+x))/(1-x+sqrt(1-2*x-3*x^2))^2.

a(n) = (1/n) * Sum_{j=0..n} ((binomial(j,n-1-j)+4*binomial(j,n-2-j) + 3*binomial(j,n-3-j))*binomial(n,j)). - Vladimir Kruchinin, Mar 08 2016

a(n) ~ 4*3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 08 2016

a(n) = A001006(n+1) - A001006(n-1). - Gennady Eremin, Sep 23 2021

D-finite with recurrence (n+3)*a(n) + (-3*n-5)*a(n-1) + (-n+3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 01 2021

From Peter Bala, Feb 02 2024: (Start)

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A002057(k).

G.f.: x/(1 + x)*c(x/(1 + x))^4, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)