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
Examples
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;
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)
Programs
-
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
-
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
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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 -
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
Formula
Sum_{k=0..n} T(n, k) = A004123(n+2).
Extensions
Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010