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%I #15 Jan 06 2019 06:08:26
%S 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 320,300,204,65,129,462,756,840,840,756,462,129,257,1040,1904,2240,
%U 2240,2240,1904,1040,257,513,2322,4752,6048,6048,6048,6048,4752,2322,513
%N Triangle read by rows: T(n,m) = (2^(n-m) + 2^m)*binomial(n,m), 0 <= m <= n.
%C Row sums are A025192(n+1).
%H A. Lakhtakia, R. Messier, V. K. Varadan, 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 T(n,m) = A007318(n,m)*(2^(n-m) + 2^m).
%e 2;
%e 3, 3;
%e 5, 8, 5;
%e 9, 18, 18, 9;
%e 17, 40, 48, 40, 17;
%e 33, 90, 120, 120, 90, 33;
%e 65, 204, 300, 320, 300, 204, 65;
%e 129, 462, 756, 840, 840, 756, 462, 129;
%e 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257;
%e 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513;
%e 1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;
%p A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # _R. J. Mathar_, Jan 13 2011
%t t[n_, m_] := (2^(n - m) + 2^m) Binomial[n, m]; Table[ t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten
%t (* alternate program *)
%t Table[Table[ Sum[Binomial[n, m]*Binomial[m, k] + Binomial[n, n - m]*Binomial[m, m - k], {k, 0, n}]/2, {m, 0, n}]
%t + Reverse[ Table[Sum[ Binomial[n, m]*Binomial[m, k] + Binomial[n, n - m]*Binomial[m, m - k], {k, 0, n}]/2, {m, 0, n}]], {n, 0, 10}] (* _Roger L. Bagula_, Oct 14 2010 *)
%Y Cf. A025192.
%K nonn,tabl,easy
%O 0,1
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009
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