A154693 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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)

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)

FORMULA

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

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

EXAMPLE

The triangle begins as:

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;

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

PROG

(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

(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

CROSSREFS

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

Sequence in context: A045626 A275914 A154923 * A065854 A263769 A064776

Adjacent sequences: A154690 A154691 A154692 * A154694 A154695 A154696

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

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

STATUS

approved