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.

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 247, 65, 1, 1, 211, 2743, 2743, 211, 1, 1, 665, 28063, 96005, 28063, 665, 1, 1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1, 1, 6305, 2596399, 90873965, 294990871, 90873965, 2596399, 6305, 1, 1, 19171, 24174631, 2619766591, 26802227431, 26802227431, 2619766591, 24174631, 19171, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = 2^(k*(n-k)) * q-binomial(n, k, 3/2).

T(n,k) = 3^(n-k) * T(n-1,k-1) + 2^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) = 3^j - 2^j.

Example

Triangle begins:
  1;
  1,    1;
  1,    5,      1;
  1,   19,     19,       1;
  1,   65,    247,      65,       1;
  1,  211,   2743,    2743,     211,      1;
  1,  665,  28063,   96005,   28063,    665,    1;
  1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1;
  ...

Prog

(PARI) T(n, k) = if(n*k==0, n^k, 2^(n-k)*T(n-1, k-1)+3^k*T(n-1, k));
(SageMath)
def a_row(n): return [2^(k*(n-k))*q_binomial(n, k, 3/2) for k in (0..n)]
for n in (0..9): print(a_row(n))

Crossrefs

Columns k=0..3 give A000012, A001047, A019443(n-2), A383754(n-3).

Cf. A022167.

Sequence in context: A395490 A168551 A262307 * A144397 A047909 A395149

Adjacent sequences: A383750 A383751 A383752 * A383754 A383755 A383756

Keyword

nonn,tabl

Author

Seiichi Manyama, May 09 2025

Status

approved