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%I #29 Sep 08 2022 08:45:50
%S 1,9,56,297,1444,6656,29618,128603,548591,2309467,9624964,39799813,
%T 163556776,668796712,2723729944,11055878188,44753742226,180746332690,
%U 728571706240,2932018571370,11783070278816,47297147250204
%N Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=8.
%C This sequence is the 8th 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.
%H Vincenzo Librandi, <a href="/A172065/b172065.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{j=0..n} (-1)^j *binomial(2*n+k-j, n-j), with k=8.
%F a(n) ~ 2^(2*n+9)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 19 2014
%F Conjecture: 2*n*(n+8)*(3*n+13)*a(n) -(21*n^3 + 247*n^2 + 980*n + 1344)*a(n-1) - 2*(n+3)*(3*n+16)*(2*n+7)*a(n-2) = 0. - _R. J. Mathar_, Feb 29 2016
%e a(4) = C(16,4) - C(15,3) + C(14,2) - C(13,1) + C(12,0) = 20*91 - 35*13 + 91 - 13 + 1 = 1820 - 455 + 79 = 1444.
%p a:= n-> add((-1)^(p)*binomial(2*n-p+8, n-p), p=0..n):
%p seq(a(n), n=0..40);
%p # 2nd program
%p a:= n-> coeff(series((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))
%p /(2*z))^8, z, n+20), z, n):
%p seq(a(n), n=0..40);
%t CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^8, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 19 2014 *)
%o (PARI) k=8; 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 17 2019
%o (Magma) k:=8 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 17 2019
%o (Sage) k=8; m=30; a=((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k ).series(x, m+2).coefficients(x, sparse=False); a[0:m] # _G. C. Greubel_, Feb 17 2019
%Y Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172066 (k=9), A172067 (k=10)
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Jan 24 2010