A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

2, 3, 3, 5, 16, 5, 9, 66, 66, 9, 17, 260, 528, 260, 17, 33, 1026, 3624, 3624, 1026, 33, 65, 4080, 23820, 38656, 23820, 4080, 65, 129, 16302, 154548, 374856, 374856, 154548, 16302, 129, 257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260, 257

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

Keywords

• nonn
• tabl

Comments

From G. C. Greubel, Jan 17 2025: (Start)

A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A008292(n+1, k+1). When (p, q) = (2, 1) this sequence is obtained.

Some related triangles are:

(p, q) = (1, 1) : 2*A008292(n,k).

(p, q) = (2, 2) : 2*A257609(n,k).

(p, q) = (3, 2) : A154694(n,k).

(p, q) = (3, 3) : 2*A257620(n,k). (End)

Examples

			The triangle begins as:
    2;
    3,     3;
    5,    16,      5;
    9,    66,     66,       9;
   17,   260,    528,     260,      17;
   33,  1026,   3624,    3624,    1026,      33;
   65,  4080,  23820,   38656,   23820,    4080,     65;
  129, 16302, 154548,  374856,  374856,  154548,  16302,   129;
  257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260,  257;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2501, (FIG. 3)

Crossrefs

Cf. A000629 (row sums), A008292, A154694, A257609, A257620.

Programs

Magma

A154693:= func< n,k | (2^(n-k) + 2^k)*EulerianNumber(n+1, k) >;
[A154693(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025

Mathematica

p=2; q=1;
A008292[n_,k_]:= A008292[n,k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
T[n_, m_]:= (p^(n-m)*q^m + p^m*q^(n-m))*A008292[n+1,m+1];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

SageMath

from sage.combinat.combinat import eulerian_number
def A154693(n,k): return (2^(n-k) +2^k)*eulerian_number(n+1,k)
print(flatten([[A154693(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

Formula

T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1)

Sum_{k=0..n} T(n, k) = A000629(n+1).

Extensions

Definition simplified by the Assoc. Eds. of the OEIS - Aug 08 2010.