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 A154690 #25 Jan 23 2025 00:18:09
%S A154690 2,3,3,5,8,5,9,18,18,9,17,40,48,40,17,33,90,120,120,90,33,65,204,300,
%T A154690 320,300,204,65,129,462,756,840,840,756,462,129,257,1040,1904,2240,
%U A154690 2240,2240,1904,1040,257,513,2322,4752,6048,6048,6048,6048,4752,2322,513
%N A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.
%C A154690 From _G. C. Greubel_, Jan 18 2025: (Start)
%C A154690 A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
%C A154690 Some related triangles are:
%C A154690 (p, q) = (1, 1) : 2*A007318(n,k).
%C A154690 (p, q) = (2, 2) : 2*A038208(n,k).
%C A154690 (p, q) = (3, 2) : A154692(n,k).
%C A154690 (p, q) = (3, 3) : 2*A038221(n,k). (End)
%H A154690 G. C. Greubel, <a href="/A154690/b154690.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A154690 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 A154690 T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
%F A154690 Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
%F A154690 From _G. C. Greubel_, Jan 18 2025: (Start)
%F A154690 T(n, n-k) = T(n, k) (symmetry).
%F A154690 T(n, 1) = n + A215149(n), n >= 1.
%F A154690 T(2*n-1, n) = 3*A069720(n).
%F A154690 Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
%F A154690 Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
%F A154690 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)
%e A154690 Triangle begins as:
%e A154690 2;
%e A154690 3, 3;
%e A154690 5, 8, 5;
%e A154690 9, 18, 18, 9;
%e A154690 17, 40, 48, 40, 17;
%e A154690 33, 90, 120, 120, 90, 33;
%e A154690 65, 204, 300, 320, 300, 204, 65;
%e A154690 129, 462, 756, 840, 840, 756, 462, 129;
%e A154690 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257;
%e A154690 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513;
%e A154690 1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;
%p A154690 A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # _R. J. Mathar_, Jan 13 2011
%t A154690 T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
%t A154690 Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten
%o A154690 (Magma)
%o A154690 A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
%o A154690 [A154690(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 18 2025
%o A154690 (Python)
%o A154690 from sage.all import *
%o A154690 def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
%o A154690 print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 18 2025
%Y A154690 Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
%Y A154690 Cf. A215149.
%Y A154690 Sums include: A008776 (row), A010673 (alternating sign row).
%Y A154690 Columns k: A000051 (k=0).
%Y A154690 Main diagonal: A059304.
%K A154690 nonn,tabl,easy
%O A154690 0,1
%A A154690 _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009