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 A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4. 13
 1, 5, 22, 91, 367, 1461, 5776, 22748, 89402, 350974, 1377174, 5403193, 21201211, 83211277, 326703424, 1283211208, 5042294926, 19822108582, 77958648604, 306739666198, 1207433301046, 4754874514690, 18732340230592, 73827134976216 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is the 4th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array. Apparently the number of peaks in all Dyck paths of semilength n+4 that are 2 steps higher than the preceding peak. - David Scambler, Apr 22 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=4. a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p, n-p), with k=4. a(n) ~ 2^(2*n+5)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014 D-finite with recurrence: +2*(n+4)*a(n) +(-13*n-36)*a(n-1) +(15*n+16)*a(n-2) +(19*n+14)*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Feb 21 2020 EXAMPLE a(4) = C(12,4)-C(11,3)+C(10,2)-C(9,1)+C(8,0)=55*9-55*3+45-9+1=367. MAPLE a:= n-> add((-1)^(p)*binomial(2*n+4-p, n-p), p=0..n): seq(a(n), n=0..30); # second Maple program: gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^4: a:= n-> coeff(series(gf, z, n+10), z, n): seq(a(n), n=0..30); MATHEMATICA CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *) PROG (PARI) k=4; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 16 2019 (MAGMA) k:=4; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 16 2019 (Sage) k=4; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 16 2019 CROSSREFS Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10). Sequence in context: A050185 A216597 A085812 * A211973 A053297 A071715 Adjacent sequences:  A172058 A172059 A172060 * A172062 A172063 A172064 KEYWORD easy,nonn AUTHOR Richard Choulet, Jan 24 2010 STATUS approved

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Last modified December 1 03:31 EST 2020. Contains 338833 sequences. (Running on oeis4.)