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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.

%I A172025 #40 Sep 08 2022 08:45:50
%S A172025 1,4,16,62,239,920,3544,13672,52834,204528,793092,3080226,11980667,
%T A172025 46662704,181971248,710454896,2776717742,10863073784,42537035408,
%U A172025 166704021596,653827252022,2566222449104,10079023179536,39611016586832
%N A172025 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=3.
%C A172025 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:
%C A172025   1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,
%C A172025   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
%C A172025   1,  1,  2,  2,  3,  3,  4,  4,  5,  5,  6,  6,  7,  7,
%C A172025   1,  2,  4,  6,  9, 12, 16, 20, 25, 30,
%C A172025   1,  3,  7, 13, 22, 34, 50, 70, 95.
%C A172025 The Maple programs give the first diagonals of this array.
%C A172025 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
%H A172025 Vincenzo Librandi, <a href="/A172025/b172025.txt">Table of n, a(n) for n = 0..1000</a>
%F A172025 G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=3.
%F A172025 a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p,n-p), with k=3.
%F A172025 a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 19 2014
%F A172025 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
%F A172025 a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(n+3)). - _Ilya Gutkovskiy_, Oct 25 2017
%e A172025 a(4) = C(11,4) - C(10,3) + C(9,2) - C(8,1) + C(7,0) = 330 - 120 + 36 - 8 + 1 = 239.
%p A172025 a:= n-> add((-1)^(p)*binomial(2*n+3-p,n-p), p=0..n):
%p A172025 seq(a(n), n=0..30);
%p A172025 # second Maple program:
%p A172025 gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^3:
%p A172025 a:= n-> coeff(series(gf,z,n+10),z,n):
%p A172025 seq(a(n), n=0..30);
%t A172025 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 *)
%o A172025 (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
%o A172025 (Magma) k:=3; m:=30; R<x>:=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
%o A172025 (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
%Y A172025 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).
%K A172025 easy,nonn
%O A172025 0,2
%A A172025 _Richard Choulet_, Jan 23 2010