A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Author

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

Keywords

• nonn
• tabl
• easy

Comments

From G. C. Greubel, Jan 18 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))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.

Some related triangles are:

(p, q) = (1, 1) : 2*A007318(n,k).

(p, q) = (2, 2) : 2*A038208(n,k).

(p, q) = (3, 2) : A154692(n,k).

(p, q) = (3, 3) : 2*A038221(n,k). (End)

Examples

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;
		

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. 2502, Fig. 3.

Crossrefs

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.

Cf. A215149.

Sums include: A008776 (row), A010673 (alternating sign row).

Columns k: A000051 (k=0).

Main diagonal: A059304.

Programs

Magma

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

Maple

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

Mathematica

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

Python

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Formula

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).

Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).

From G. C. Greubel, Jan 18 2025: (Start)

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

T(n, 1) = n + A215149(n), n >= 1.

T(2*n-1, n) = 3*A069720(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)