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 A383119 #12 Apr 17 2025 08:11:03
%S A383119 1,3,21,147,1093,8343,64869,510891,4062277,32539647,262181601,
%T A383119 2122581123,17252278789,140695104943,1150670390541,9433965332127,
%U A383119 77512716483461,638080242074447,5261486780929209,43450477494413751,359308411992366513,2974886601163646379,24657831769475675253
%N A383119 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).
%C A383119 Inverse binomial transform of A005810.
%F A383119 G.f.: (1/x) * Sum_{k>=0} binomial(4*k,k) * (x/(1 + x))^(k+1).
%F A383119 a(n) = [x^n] (1 + 3*x + 6*x^2 + 4*x^3 + x^4)^n.
%F A383119 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x/((1+x)^4 - x) ). See A317133. - _Seiichi Manyama_, Apr 17 2025
%F A383119 a(n) ~ 229^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)). - _Vaclav Kotesovec_, Apr 17 2025
%t A383119 Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k], {k, 0, n}], {n, 0, 22}]
%t A383119 Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {1/3, 2/3, 1}, 256/27], {n, 0, 22}]
%t A383119 nmax = 22; CoefficientList[Series[(1/x) Sum[Binomial[4 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%o A383119 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(4*k, k)); \\ _Seiichi Manyama_, Apr 17 2025
%Y A383119 Cf. A002426, A005810, A346664, A359643, A383118.
%Y A383119 Cf. A317133.
%K A383119 nonn
%O A383119 0,2
%A A383119 _Ilya Gutkovskiy_, Apr 17 2025