A154694 Triangle read by rows: T(n,k) = ((3/2)^k*2^n + (2/3)^k*3^n)*A008292(n+1,k+1).

2, 5, 5, 13, 48, 13, 35, 330, 330, 35, 97, 2028, 4752, 2028, 97, 275, 11970, 54360, 54360, 11970, 275, 793, 69840, 557388, 1043712, 557388, 69840, 793, 2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315, 6817, 2388516, 51011136, 247761072, 404844480, 247761072, 51011136, 2388516, 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 34 (3) (1986) 1986, page 2502, (FIG. 3)

Formula

Sum_{k=0..n} T(n, k) = A004123(n+2).

Example

Triangle begins as:
     2;
     5,      5;
    13,     48,      13;
    35,    330,     330,       35;
    97,   2028,    4752,     2028,       97;
   275,  11970,   54360,    54360,    11970,     275;
   793,  69840,  557388,  1043712,   557388,   69840,    793;
  2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315;

Maple

A154694 := proc(n, m)
    (3^m*2^(n-m)+2^m*3^(n-m))*A008292(n+1, m+1) ;
end proc:
seq(seq( A154694(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Mar 11 2024

Mathematica

T[n_, k_, p_, q_] := (p^(n - k)*q^k + p^k*q^(n - k))*Eulerian[n+1, k];
Table[T[n, k, 2, 3], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
A154694:= func< n, k | (2^(n-k)*3^k+2^k*3^(n-k))*EulerianNumber(n+1, k) >;
[A154694(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025
(Python)
from sage.all import *
from sage.combinat.combinat import eulerian_number
def A154694(n, k): return (pow(2, n-k)*pow(3, k)+pow(2, k)*pow(3, n-k))*eulerian_number(n+1, k)
print(flatten([[A154694(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Crossrefs

Cf. A004123 (row sums), A154693, A256890.

Sequence in context: A174098 A183419 A305314 * A154696 A154698 A063786

Adjacent sequences: A154691 A154692 A154693 * A154695 A154696 A154697

Keyword

nonn,tabl,easy

Author

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

Extensions

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

Status

approved