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 A375181 #22 Oct 24 2024 05:06:43
%S A375181 1,1,2,3,6,12,28,70,189,533,1551,4599,13807,41784,127144,388373,
%T A375181 1189661,3651910,11228851,34571914,106555917,328713138,1014797705,
%U A375181 3134841053,9689148780,29961083746,92683964271,286816872102,887849075464,2749110140301,8514323952447
%N A375181 Inverse binomial transform of A376277.
%C A375181 The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
%C A375181 The Hankel sequence transform starting with the second term gives {1, -3, -2, 1, 3, 2, -1, -3, -2, ...}.
%C A375181 The periodic pattern in the continued fractions of the generating function starts after some prefix, this causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.
%F A375181 G.f.: 1/(1-1*x/(1-1*x/(1+1*x/(1+1*x/(1-3*x/(1-(1/3)*x/(1-(2/3)*x/(1-(3/2)*x/(1+(1/2)*x/(...)))))))))), a continued fraction expansion. The coefficients of x are {-1, -1, 1, 1, -3, -(1/3), -(2/3), -(3/2), (1/2), 2, -3, ...}. The numerators will repeat {1, 2, 3} the denominators {1, 1, 2, 2, 3, 3} the sign repeats {-,-,-,-,+,+}.
%F A375181 G.f.: (2*x^2 - sqrt(-3*x^2 - 2*x + 1) + 3*x - 1)/(-5*x^3 + sqrt(-3*x^2 - 2*x + 1)*(x^2 + x - 1) + 4*x - 1)
%F A375181 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A376277(k).
%F A375181 D-finite with recurrence (-n+3)*a(n) +(7*n-25)*a(n-1) +2*(-6*n+25)*a(n-2) +10*(-n+4)*a(n-3) +2*(16*n-73)*a(n-4) +(n-5)*a(n-5) +21*(-n+5)*a(n-6)=0. - _R. J. Mathar_, Oct 24 2024
%o A375181 (PARI) my(N=30, x='x+O('x^N)); Vec((2*x^2-sqrt(-3*x^2-2*x+1)+3*x-1)/(-5*x^3+sqrt(-3*x^2-2*x+1)*(x^2+x-1)+4*x-1))
%Y A375181 Cf. A059738, A376277.
%K A375181 nonn,less
%O A375181 0,3
%A A375181 _Thomas Scheuerle_, Sep 23 2024