This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A154694 #11 Jan 18 2025 03:10:24
%S A154694 2,5,5,13,48,13,35,330,330,35,97,2028,4752,2028,97,275,11970,54360,
%T A154694 54360,11970,275,793,69840,557388,1043712,557388,69840,793,2315,
%U A154694 407550,5409180,16868520,16868520,5409180,407550,2315,6817,2388516,51011136,247761072,404844480,247761072,51011136,2388516,6817
%N A154694 Triangle read by rows: T(n,k) = ((3/2)^k*2^n + (2/3)^k*3^n)*A008292(n+1,k+1).
%H A154694 G. C. Greubel, <a href="/A154694/b154694.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A154694 A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, <a href="http://dx.doi.org/10.1103/PhysRevA.34.2501">Use of combinatorial algebra for diffusion on fractals</a>, Physical Review A 34 (3) (1986) 1986, page 2502, (FIG. 3)
%F A154694 Sum_{k=0..n} T(n, k) = A004123(n+2).
%e A154694 Triangle begins as:
%e A154694 2;
%e A154694 5, 5;
%e A154694 13, 48, 13;
%e A154694 35, 330, 330, 35;
%e A154694 97, 2028, 4752, 2028, 97;
%e A154694 275, 11970, 54360, 54360, 11970, 275;
%e A154694 793, 69840, 557388, 1043712, 557388, 69840, 793;
%e A154694 2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315;
%p A154694 A154694 := proc(n,m)
%p A154694 (3^m*2^(n-m)+2^m*3^(n-m))*A008292(n+1,m+1) ;
%p A154694 end proc:
%p A154694 seq(seq( A154694(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Mar 11 2024
%t A154694 T[n_, k_, p_, q_] := (p^(n - k)*q^k + p^k*q^(n - k))*Eulerian[n+1,k];
%t A154694 Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten
%o A154694 (Magma)
%o A154694 A154694:= func< n,k | (2^(n-k)*3^k+2^k*3^(n-k))*EulerianNumber(n+1, k) >;
%o A154694 [A154694(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 18 2025
%o A154694 (Python)
%o A154694 from sage.all import *
%o A154694 from sage.combinat.combinat import eulerian_number
%o A154694 def A154694(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*eulerian_number(n+1,k)
%o A154694 print(flatten([[A154694(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 18 2025
%Y A154694 Cf. A004123 (row sums), A154693, A256890.
%K A154694 nonn,tabl,easy
%O A154694 0,1
%A A154694 _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009
%E A154694 Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010