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 A349290 #23 Jul 10 2025 22:40:54
%S A349290 1,2,11,96,1001,11456,139013,1756596,22867421,304560171,4130200726,
%T A349290 56836946342,791689962811,11140615233281,158140107648676,
%U A349290 2261708608884896,32559326010349817,471428798399646336,6860801662510005266,100302910051255600486
%N A349290 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^4)).
%H A349290 Seiichi Manyama, <a href="/A349290/b349290.txt">Table of n, a(n) for n = 0..500</a>
%F A349290 a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
%F A349290 a(n) = F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], -3^3*5^5/2^16), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 13 2021
%F A349290 a(n) ~ sqrt(1 + 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1-r)) * n^(3/2) * r^(n + 1/4)), where r = 0.0631152861998150860738633360987635931... is the root of the equation 5^5 * r = 4^4 * (1-r)^4. - _Vaclav Kotesovec_, Nov 14 2021
%F A349290 a(n) = 1 + Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - _Seiichi Manyama_, Jul 10 2025
%t A349290 nmax = 19; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349290 Table[Sum[Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]
%o A349290 (PARI) a(n) = sum(k=0, n, binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ _Michel Marcus_, Nov 14 2021
%Y A349290 Cf. A002294, A007317, A199475, A346647, A349289, A349291, A349292, A349293.
%K A349290 nonn
%O A349290 0,2
%A A349290 _Ilya Gutkovskiy_, Nov 13 2021