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A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

%I A383755 #20 May 09 2025 11:42:07
%S A383755 1,1,1,1,7,1,1,37,37,1,1,175,925,175,1,1,781,19525,19525,781,1,1,3367,
%T A383755 375661,1776775,375661,3367,1,1,14197,6828757,144142141,144142141,
%U A383755 6828757,14197,1,1,58975,119609725,10884484975,48575901517,10884484975,119609725,58975,1
%N A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.
%F A383755 T(n,k) = 3^(k*(n-k)) * q-binomial(n, k, 4/3).
%F A383755 T(n,k) = 4^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k).
%F A383755 T(n,k) = T(n,n-k).
%F A383755 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 3^j.
%e A383755 Triangle begins:
%e A383755   1;
%e A383755   1,    1;
%e A383755   1,    7,      1;
%e A383755   1,   37,     37,       1;
%e A383755   1,  175,    925,     175,      1;
%e A383755   1,  781,  19525,   19525,    781,    1;
%e A383755   1, 3367, 375661, 1776775, 375661, 3367, 1;
%e A383755   ...
%o A383755 (PARI) T(n, k) = if(n*k==0, n^k, 3^(n-k)*T(n-1, k-1)+4^k*T(n-1, k));
%o A383755 (Sage)
%o A383755 def a_row(n): return [3^(k*(n-k))*q_binomial(n, k, 4/3) for k in (0..n)]
%o A383755 for n in (0..8): print(a_row(n))
%Y A383755 Columns k=0..3 give A000012, A005061, A383756(n-2), A383757(n-3).
%Y A383755 Cf. A022168.
%K A383755 nonn,tabl
%O A383755 0,5
%A A383755 _Seiichi Manyama_, May 09 2025