This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A172066 #20 Sep 08 2022 08:45:50
%S A172066 1,10,67,376,1912,9142,41941,186880,815083,3498146,14827487,62236064,
%T A172066 259187048,1072567256,4415408372,18098359424,73915594466,300958990724,
%U A172066 1222228100590,4952609171080,20030298812596,80876902778482
%N A172066 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=9.
%C A172066 This sequence is the 9th 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 A172066 Vincenzo Librandi, <a href="/A172066/b172066.txt">Table of n, a(n) for n = 0..200</a>
%F A172066 a(n) = Sum_{j=0..n} (-1)^j * binomial(2*n+k-j,n-j), with k=9.
%F A172066 a(n) ~ 2^(2*n+10)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 19 2014
%F A172066 Conjecture: 2*n*(n+9)*(n+5)*a(n) -(7*n^3+94*n^2+427*n+672)*a(n-1) -2*(2*n+7)*(n+6)*(n+4)*a(n-2)=0. - _R. J. Mathar_, Feb 19 2016
%e A172066 a(4) = C(17,4) - C(16,3) + C(15,2) - C(14,1) + C(13,0) = 17*4*5*7 - 16*5*7 + 105 - 14 + 1 = 5*7*(68-16) + 92 = 1912.
%p A172066 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od;
%p A172066 # 2nd program
%p A172066 for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%t A172066 CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^9, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 19 2014 *)
%o A172066 (PARI) k=9; 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 A172066 (Magma) k:=9; 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 A172066 (Sage) k=9; 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 A172066 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), A172065 (k=8), A172067 (k=10).
%K A172066 easy,nonn
%O A172066 0,2
%A A172066 _Richard Choulet_, Jan 24 2010