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 A383753 #24 May 09 2025 11:42:14
%S A383753 1,1,1,1,5,1,1,19,19,1,1,65,247,65,1,1,211,2743,2743,211,1,1,665,
%T A383753 28063,96005,28063,665,1,1,2059,273847,3041143,3041143,273847,2059,1,
%U A383753 1,6305,2596399,90873965,294990871,90873965,2596399,6305,1,1,19171,24174631,2619766591,26802227431,26802227431,2619766591,24174631,19171,1
%N A383753 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 2^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k) with T(n,k) = n^k if n*k=0.
%F A383753 T(n,k) = 2^(k*(n-k)) * q-binomial(n, k, 3/2).
%F A383753 T(n,k) = 3^(n-k) * T(n-1,k-1) + 2^k * T(n-1,k).
%F A383753 T(n,k) = T(n,n-k).
%F A383753 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 2^j.
%e A383753 Triangle begins:
%e A383753 1;
%e A383753 1, 1;
%e A383753 1, 5, 1;
%e A383753 1, 19, 19, 1;
%e A383753 1, 65, 247, 65, 1;
%e A383753 1, 211, 2743, 2743, 211, 1;
%e A383753 1, 665, 28063, 96005, 28063, 665, 1;
%e A383753 1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1;
%e A383753 ...
%o A383753 (PARI) T(n, k) = if(n*k==0, n^k, 2^(n-k)*T(n-1, k-1)+3^k*T(n-1, k));
%o A383753 (Sage)
%o A383753 def a_row(n): return [2^(k*(n-k))*q_binomial(n, k, 3/2) for k in (0..n)]
%o A383753 for n in (0..9): print(a_row(n))
%Y A383753 Columns k=0..3 give A000012, A001047, A019443(n-2), A383754(n-3).
%Y A383753 Cf. A022167.
%K A383753 nonn,tabl
%O A383753 0,5
%A A383753 _Seiichi Manyama_, May 09 2025