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A025235
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a(n) = (1/2)*s(n+2), where s = A014431.
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3
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1, 1, 3, 7, 21, 61, 191, 603, 1961, 6457, 21595, 72975, 249085, 857013, 2970007, 10356323, 36311633, 127937649, 452738867, 1608426647, 5734534629, 20511509549, 73583105007, 264687136235, 954482676217, 3449853902761, 12495597328011, 45349353908383
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| G.f. (offset 1) is series reversion of x/(1+x+2x^2). - Michael Somos, Jul 12 2003.
Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in two colors: red (R) and green (G) (i.e. Motzkin paths with the up steps in two colors). E.g. a(3)=7 because we have HHH, HRD, HGD, RDH, GDH, RHD and GHD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2003
Equals inverse binomial transform of A071356: (1, 2, 6, 20, 72,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 03 2010]
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REFERENCES
| Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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FORMULA
| a(n)=sum(2^(k-1)*binomial(n+1, k)*binomial(n-k+1, k-1), k=0..n)/(n+1) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
G.f.: (1-x-sqrt(1-2x-7x^2))/(4x^2). - Michael Somos, Jun 08 2000.
a(n)=sum{k=0..n, binomial(n, k)2^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A00108(n). - Paul Barry (pbarry(AT)wit.ie), Dec 22 2003
E.g.f.: exp(x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 31 2004
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x/(1+x+2*x^2+x*O(x^n))), n+1))
(PARI) {a(n)= if(n<0, 0, polcoeff( (1-x- sqrt(1 -2*x -7*x^2 +x^3*O(x^n)) )/4, n+2))} /* Michael Somos Mar 31 2007 */
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); n!*simplify(polcoeff( exp(x +A)* besseli(1, 2*x* quadgen(8) +A), n)))} /* Michael Somos Mar 31 2007 */
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CROSSREFS
| Cf. A071356, A001003, A068764.
A071356 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 03 2010]
Sequence in context: A102877 A122983 A005355 * A129366 A166358 A148670
Adjacent sequences: A025232 A025233 A025234 * A025236 A025237 A025238
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KEYWORD
| nonn,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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