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.

1, 1, 1, 1, 7, 1, 1, 37, 37, 1, 1, 175, 925, 175, 1, 1, 781, 19525, 19525, 781, 1, 1, 3367, 375661, 1776775, 375661, 3367, 1, 1, 14197, 6828757, 144142141, 144142141, 6828757, 14197, 1, 1, 58975, 119609725, 10884484975, 48575901517, 10884484975, 119609725, 58975, 1

Offset: 0

Author

Seiichi Manyama, May 09 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,    1;
  1,    7,      1;
  1,   37,     37,       1;
  1,  175,    925,     175,      1;
  1,  781,  19525,   19525,    781,    1;
  1, 3367, 375661, 1776775, 375661, 3367, 1;
  ...
		

Crossrefs

Columns k=0..3 give A000012, A005061, A383756(n-2), A383757(n-3).

Cf. A022168.

Programs

PARI

T(n, k) = if(n*k==0, n^k, 3^(n-k)*T(n-1, k-1)+4^k*T(n-1, k));

Sage

def a_row(n): return [3^(k*(n-k))*q_binomial(n, k, 4/3) for k in (0..n)]
for n in (0..8): print(a_row(n))

Formula

T(n,k) = 3^(k*(n-k)) * q-binomial(n, k, 4/3).

T(n,k) = 4^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k).

T(n,k) = T(n,n-k).

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.