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A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

%I A154693 #16 Jan 18 2025 09:08:23
%S A154693 2,3,3,5,16,5,9,66,66,9,17,260,528,260,17,33,1026,3624,3624,1026,33,
%T A154693 65,4080,23820,38656,23820,4080,65,129,16302,154548,374856,374856,
%U A154693 154548,16302,129,257,65260,993344,3529360,4998080,3529360,993344,65260,257
%N A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).
%C A154693 From _G. C. Greubel_, Jan 17 2025: (Start)
%C A154693 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))*A008292(n+1, k+1). When (p, q) = (2, 1) this sequence is obtained.
%C A154693 Some related triangles are:
%C A154693   (p, q) = (1, 1) : 2*A008292(n,k).
%C A154693   (p, q) = (2, 2) : 2*A257609(n,k).
%C A154693   (p, q) = (3, 2) : A154694(n,k).
%C A154693   (p, q) = (3, 3) : 2*A257620(n,k). (End)
%H A154693 G. C. Greubel, <a href="/A154693/b154693.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A154693 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. 2501, (FIG. 3)
%F A154693 T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1)
%F A154693 Sum_{k=0..n} T(n, k) = A000629(n+1).
%e A154693 The triangle begins as:
%e A154693     2;
%e A154693     3,     3;
%e A154693     5,    16,      5;
%e A154693     9,    66,     66,       9;
%e A154693    17,   260,    528,     260,      17;
%e A154693    33,  1026,   3624,    3624,    1026,      33;
%e A154693    65,  4080,  23820,   38656,   23820,    4080,     65;
%e A154693   129, 16302, 154548,  374856,  374856,  154548,  16302,   129;
%e A154693   257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260,  257;
%t A154693 p=2; q=1;
%t A154693 A008292[n_,k_]:= A008292[n,k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
%t A154693 T[n_, m_]:= (p^(n-m)*q^m + p^m*q^(n-m))*A008292[n+1,m+1];
%t A154693 Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten
%o A154693 (Magma)
%o A154693 A154693:= func< n,k | (2^(n-k) + 2^k)*EulerianNumber(n+1, k) >;
%o A154693 [A154693(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 17 2025
%o A154693 (SageMath)
%o A154693 from sage.combinat.combinat import eulerian_number
%o A154693 def A154693(n,k): return (2^(n-k) +2^k)*eulerian_number(n+1,k)
%o A154693 print(flatten([[A154693(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 17 2025
%Y A154693 Cf. A000629 (row sums), A008292, A154694, A257609, A257620.
%K A154693 nonn,tabl
%O A154693 0,1
%A A154693 _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009
%E A154693 Definition simplified by the Assoc. Eds. of the OEIS - Aug 08 2010.