A154692 Triangle read by rows: T(n, k) = (2^(n-k)*3^k + 2^k*3^(n-k))*binomial(n, k).

2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817

Offset

0,1

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.

Formula

Sum_{k=0..n} T(n, k) = A020729(n) = A020699(n+1).

T(n,m) = A013620(n,m) + A013620(m,n). - R. J. Mathar, Oct 24 2011

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

T(2*n, n) = A119309(n).

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

Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1) + A007482(n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)

Example

Triangle begins
     2;
     5,     5;
    13,    24,    13;
    35,    90,    90,     35;
    97,   312,   432,    312,     97;
   275,  1050,  1800,   1800,   1050,    275;
   793,  3492,  7020,   8640,   7020,   3492,   793;
  2315, 11550, 26460,  37800,  37800,  26460, 11550,  2315;
  6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;

Maple

A154692 := proc(n, m)
        (2^(n-m)*3^m+2^m*3^(n-m))*binomial(n, m) ;
end proc:
seq(seq(A154692(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Oct 24 2011

Mathematica

p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n, m];
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten

Prog

(Magma)
A154692:= func< n, k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n, k) >;
[A154692(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025
(Python)
from sage.all import *
def A154692(n, k): return (pow(2, n-k)*pow(3, k)+pow(2, k)*pow(3, n-k))*binomial(n, k)
print(flatten([[A154692(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Crossrefs

Cf. A000225, A007482, A013620, A088137, A119309.

Sums include: A010673 (alternating sign row), A020699 (row), A020729 (row).

Related sequences: A007318, A154690,

Sequence in context: A326532 A326637 A303355 * A309161 A144293 A174098

Adjacent sequences: A154689 A154690 A154691 * A154693 A154694 A154695

Keyword

nonn,tabl,easy

Author

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

Status

approved