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A156016 Expansion of (1-x-sqrt(1-6x-3x^2))/(2x). 5
1, 3, 9, 36, 162, 783, 3969, 20817, 112023, 615033, 3431403, 19398690, 110880900, 639730305, 3720657807, 21790419444, 128398625658, 760668489729, 4528069760691, 27070491820644, 162464919528222, 978463778897637 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Transform of Catalan numbers by Riordan array ((1+x)/(1-x), x(1+x)/(1-x)^2).
LINKS
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..k+1} C(k+1,j)*C(n+k-j,n-k-j)*A000108(k).
a(n+1) = 3*A107264(n-1). - Philippe Deléham, Feb 04 2009
D-finite with recurrence: (n+1)*a(n) + 3*(-2*n+1)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x) + A(x)^2). - Ilya Gutkovskiy, Jul 01 2020
MATHEMATICA
CoefficientList[Series[(1-x-Sqrt[1-6x-3x^2])/(2x), {x, 0, 30}], x] (* Harvey P. Dale, Jul 27 2014 *)
CROSSREFS
Sequence in context: A350451 A245888 A295739 * A032314 A144352 A107895
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Feb 01 2009
STATUS
approved

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Last modified June 18 17:09 EDT 2024. Contains 373482 sequences. (Running on oeis4.)