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 A172067 #15 Jan 17 2025 15:05:07
%S A172067 1,11,79,468,2486,12323,58277,266492,1188679,5202523,22436251,
%T A172067 95630272,403770544,1691678428,7042481236,29161852240,120212658034,
%U A172067 493656394350,2020590599710,8247228533780,33579755528278,136434358356201
%N A172067 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=10.
%C A172067 This sequence is the 10th 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 A172067 G. C. Greubel, <a href="/A172067/b172067.txt">Table of n, a(n) for n = 0..1000</a>
%F A172067 a(n) = Sum_{j=0..n} (-1)^j*binomial(2*n+k-j,n-j), with k=10.
%F A172067 Conjecture: 2*n*(n+10)*(3*n+17)*a(n) - (21*n^3 + 317*n^2 + 1622*n + 2880)*a(n-1) - 2*(3*n+20)*(n+4)*(2*n+9)*a(n-2) = 0. - _R. J. Mathar_, Feb 21 2016
%e A172067 a(4) = C(18,4) - C(17,3) + C(16,2) - C(15,1) + C(14,0) = 60*51 - 680 + 120 - 15 + 1 = 2486.
%p A172067 a:= n-> add((-1)^(p)*binomial(2*n-p+10, n-p), p=0..n):
%p A172067 seq(a(n), n=0..40);
%p A172067 # 2nd program
%p A172067 a:= n-> coeff(series((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))
%p A172067 /(2*z))^10, z, n+20), z, n):
%p A172067 seq(a(n), n=0..40);
%t A172067 With[{k=10}, CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^k, {x, 0, 30}], x]] (* _G. C. Greubel_, Feb 27 2019 *)
%o A172067 (PARI) k=10; 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 27 2019
%o A172067 (Magma) k:=10; 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 27 2019
%o A172067 (Sage) k=10; 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 27 2019
%Y A172067 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), A172066 (k=9), this sequence (k=10).
%K A172067 easy,nonn
%O A172067 0,2
%A A172067 _Richard Choulet_, Jan 24 2010