cp's OEIS Frontend

A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4.

%I A172061 #28 Sep 08 2022 08:45:50
%S A172061 1,5,22,91,367,1461,5776,22748,89402,350974,1377174,5403193,21201211,
%T A172061 83211277,326703424,1283211208,5042294926,19822108582,77958648604,
%U A172061 306739666198,1207433301046,4754874514690,18732340230592,73827134976216
%N A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4.
%C A172061 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.
%C A172061 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
%H A172061 Vincenzo Librandi, <a href="/A172061/b172061.txt">Table of n, a(n) for n = 0..200</a>
%F A172061 G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=4.
%F A172061 a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p, n-p), with k=4.
%F A172061 a(n) ~ 2^(2*n+5)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 19 2014
%F A172061 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
%e A172061 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.
%p A172061 a:= n-> add((-1)^(p)*binomial(2*n+4-p, n-p), p=0..n):
%p A172061 seq(a(n), n=0..30);
%p A172061 # second Maple program:
%p A172061 gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^4:
%p A172061 a:= n-> coeff(series(gf, z, n+10), z, n):
%p A172061 seq(a(n), n=0..30);
%t A172061 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 *)
%o A172061 (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
%o A172061 (Magma) k:=4; 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 A172061 (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
%Y A172061 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).
%K A172061 easy,nonn
%O A172061 0,2
%A A172061 _Richard Choulet_, Jan 24 2010