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A102071 Pairwise sums of general ballot numbers (A002026). 4
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 (list; graph; refs; listen; history; text; internal format)
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

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. A001006.

Sequence in context: A020730 A003440 A244455 * A191627 A178778 A335596

Adjacent sequences:  A102068 A102069 A102070 * A102072 A102073 A102074

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Dec 30 2004

STATUS

approved

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Last modified January 28 14:26 EST 2022. Contains 350656 sequences. (Running on oeis4.)