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

1,2

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.

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)

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A000108, A001006, A002057.

Sequence in context: A020730 A003440 A244455 * A363142 A191627 A178778

Adjacent sequences: A102068 A102069 A102070 * A102072 A102073 A102074

Keyword

nonn,easy

Author

Ralf Stephan, Dec 30 2004

Status

approved