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 A172025 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=3. 12
 1, 4, 16, 62, 239, 920, 3544, 13672, 52834, 204528, 793092, 3080226, 11980667, 46662704, 181971248, 710454896, 2776717742, 10863073784, 42537035408, 166704021596, 653827252022, 2566222449104, 10079023179536, 39611016586832 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is the third 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+3 that are 1 step higher than the preceding peak. - David Scambler, Apr 22 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 FORMULA G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=3. a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p,n-p), with k=3. a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014 Conjecture: 2*n*(n+3)*a(n) + (-7*n^2 - 17*n - 8)*a(n-1) -2*(n+2)*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Feb 19 2016 a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017 EXAMPLE a(4) = C(11,4) - C(10,3) + C(9,2) - C(8,1) + C(7,0) = 330 - 120 + 36 - 8 + 1 = 239. MAPLE a:= n-> add((-1)^(p)*binomial(2*n+3-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))^3: a:= n-> coeff(series(gf, z, n+10), z, n): seq(a(n), n=0..30); MATHEMATICA a[n_] := Binomial[2*n+3, n+3]*Hypergeometric2F1[1, -n, -3-2*n, -1]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 17 2013 *) PROG (PARI) k=3; 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:=3; 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=3; ((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), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10). Sequence in context: A113438 A268429 A195339 * A171278 A227438 A206839 Adjacent sequences:  A172022 A172023 A172024 * A172026 A172027 A172028 KEYWORD easy,nonn AUTHOR Richard Choulet, Jan 23 2010 STATUS approved

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Last modified December 7 20:33 EST 2019. Contains 329849 sequences. (Running on oeis4.)