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 A154692 #27 Jan 20 2025 09:03:14
%S A154692 2,5,5,13,24,13,35,90,90,35,97,312,432,312,97,275,1050,1800,1800,1050,
%T A154692 275,793,3492,7020,8640,7020,3492,793,2315,11550,26460,37800,37800,
%U A154692 26460,11550,2315,6817,38064,97776,157248,181440,157248,97776,38064,6817
%N A154692 Triangle read by rows: T(n, k) = (2^(n-k)*3^k + 2^k*3^(n-k))*binomial(n, k).
%H A154692 G. C. Greubel, <a href="/A154692/b154692.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A154692 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, volume 34, Number 3 (1986) p. 2502, Fig. 3.
%F A154692 Sum_{k=0..n} T(n, k) = A020729(n) = A020699(n+1).
%F A154692 T(n,m) = A013620(n,m) + A013620(m,n). - _R. J. Mathar_, Oct 24 2011
%F A154692 From _G. C. Greubel_, Jan 18 2025: (Start)
%F A154692 T(2*n, n) = A119309(n).
%F A154692 Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
%F A154692 Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1) + A007482(n).
%F A154692 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)
%e A154692 Triangle begins
%e A154692 2;
%e A154692 5, 5;
%e A154692 13, 24, 13;
%e A154692 35, 90, 90, 35;
%e A154692 97, 312, 432, 312, 97;
%e A154692 275, 1050, 1800, 1800, 1050, 275;
%e A154692 793, 3492, 7020, 8640, 7020, 3492, 793;
%e A154692 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315;
%e A154692 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;
%p A154692 A154692 := proc(n,m)
%p A154692 (2^(n-m)*3^m+2^m*3^(n-m))*binomial(n,m) ;
%p A154692 end proc:
%p A154692 seq(seq(A154692(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Oct 24 2011
%t A154692 p=2; q=3;
%t A154692 T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n,m];
%t A154692 Table[T[n,m], {n,0,10}, {m,0,n}]//Flatten
%o A154692 (Magma)
%o A154692 A154692:= func< n,k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n,k) >;
%o A154692 [A154692(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 18 2025
%o A154692 (Python)
%o A154692 from sage.all import *
%o A154692 def A154692(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*binomial(n,k)
%o A154692 print(flatten([[A154692(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 18 2025
%Y A154692 Cf. A000225, A007482, A013620, A088137, A119309.
%Y A154692 Sums include: A010673 (alternating sign row), A020699 (row), A020729 (row).
%Y A154692 Related sequences: A007318, A154690,
%K A154692 nonn,tabl,easy
%O A154692 0,1
%A A154692 _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009