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 A386826 #11 Aug 26 2025 06:47:54
%S A386826 1,7,67,694,7459,81922,912718,10273228,116522275,1329569290,
%T A386826 15244087642,175472098996,2026521318286,23470106563924,
%U A386826 272476942589884,3169997065488664,36948020548661539,431354994430077274,5043279137171450914,59041965004582271524,692026745415822877594,8119918150063503715324
%N A386826 a(n) = Sum_{k=0..n} 3^(n-k)*C(2*n,n-k).
%C A386826 Row sums of number triangle A386825.
%C A386826 Number of strings of length 2*n defined on {0,1,2,3} that have either the same number or more 0's than the combined number of 1's, 2's and 3's.
%F A386826 a(n) = 3^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/3). - _Stefano Spezia_, Aug 05 2025
%F A386826 a(n) ~ 2^(2*n-1) * 3^(n+1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 07 2025
%F A386826 D-finite with recurrence n*a(n) +(n+1)*a(n-1) +2*(-946*n+2017)*a(n-2) +144*(286*n-821)*a(n-3) +122112*(-2*n+7)*a(n-4)=0. - _R. J. Mathar_, Aug 26 2025
%F A386826 D-finite with recurrence n*(2*n-1)*a(n) +2*(-28*n^2+20*n+9)*a(n-1) +96*(2*n+1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Aug 26 2025
%e A386826 a(3)=694 counts the strings of length 6 as follows: 540 strings with three 0's, 135 with four 0's, 18 with five 0's, and 1 string with six 0's. Hence 694 = 540 + 135 + 18 + 1, where the summands come from the triangle in A386825.
%t A386826 Table[Sum[3^(n-k) Binomial[2n, n-k], {k, 0, n}], {n, 0, 21}]
%Y A386826 Cf. A128418, A386825.
%K A386826 nonn,changed
%O A386826 0,2
%A A386826 _Enrique Navarrete_, Aug 04 2025