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A168051 Expansion of (1+x+sqrt(1-2x-3x^2))/2. 5
1, 0, -1, -1, -2, -4, -9, -21, -51, -127, -323, -835, -2188, -5798, -15511, -41835, -113634, -310572, -853467, -2356779, -6536382, -18199284, -50852019, -142547559, -400763223, -1129760415, -3192727797, -9043402501, -25669818476 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.
LINKS
FORMULA
D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - R. J. Mathar, Dec 20 2011 [Edited by Michael Somos, Jan 25 2014]
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014
G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - Michael Somos, Mar 27 2014
Convolution inverse of A005043. - Michael Somos, Mar 27 2014
a(n) ~ -3^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 05 2018
From Gennady Eremin, Feb 25 2021: (Start)
For n > 1, a(n) = A167022(n) / 2.
G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)
EXAMPLE
G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* Michael Somos, Jan 25 2014 *)
PROG
(PARI) {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* Michael Somos, Jan 25 2014 */
CROSSREFS
Sequence in context: A094286 A094287 A094288 * A166587 A292440 A168049
KEYWORD
easy,sign
AUTHOR
Paul Barry, Nov 17 2009
STATUS
approved

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Last modified June 21 06:38 EDT 2024. Contains 373540 sequences. (Running on oeis4.)