A154695 - OEIS

%I #13 Jun 03 2023 06:16:19

%S 2,3,3,5,24,5,9,138,138,9,17,760,1840,760,17,33,4266,20184,20184,4266,

%T 33,65,24548,210860,376768,210860,24548,65,129,143814,2183652,6233352,

%U 6233352,2183652,143814,129,257,851760,22549616,99411520,149600448,99411520,22549616,851760,257

%N Triangular sequence defined by T(n, m) = (r^(n-m)*q^m + r^m*q^(n-m))*b(n), where b(n) = coefficients of p(x, n) = 2^n*(1-x)^(n+1) * LerchPhi(x, -n, 1/2), and r=2, q=1.

%H G. C. Greubel, <a href="/A154695/b154695.txt">Rows n = 0..40 of triangle, flattened</a>

%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 Let r = 2 and q = 1 then b(n) = the coefficients of p(x, n) =  2^n*(1 - x)^(n + 1)* LerchPhi(x, -n, 1/2). The triangle is then defined by T(n, m) = (r^(n-m)*q^m + r^m*q^(n-m))*b(n).

%e Triangle begins as:

%e     2;

%e     3,      3;

%e     5,     24,       5;

%e     9,    138,     138,       9;

%e    17,    760,    1840,     760,      17;

%e    33,   4266,   20184,   20184,    4266,      33;

%e    65,  24548,  210860,  376768,  210860,   24548,     65;

%e   129, 143814, 2183652, 6233352, 6233352, 2183652, 143814, 129;

%t r = 2; q = 1; p[x_, n_] = 2^n*(1-x)^(n+1)*LerchPhi[x, -n, 1/2];

%t b:= Table[CoefficientList[Series[p[x, n], {x, 0, 30}], x], {n, 0, 20}];

%t T[n_, m_]:= (r^(n-m)*q^m + r^m*q^(n-m))*b[[n+1]][[m+1]];

%t Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by _G. C. Greubel_, May 08 2019 *)

%K nonn,tabl,less

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009

%E Edited by _G. C. Greubel_, May 08 2019