A154695 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.

2, 3, 3, 5, 24, 5, 9, 138, 138, 9, 17, 760, 1840, 760, 17, 33, 4266, 20184, 20184, 4266, 33, 65, 24548, 210860, 376768, 210860, 24548, 65, 129, 143814, 2183652, 6233352, 6233352, 2183652, 143814, 129, 257, 851760, 22549616, 99411520, 149600448, 99411520, 22549616, 851760, 257

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

Keywords

• nonn
• tabl
• less

Examples

			Triangle begins as:
    2;
    3,      3;
    5,     24,       5;
    9,    138,     138,       9;
   17,    760,    1840,     760,      17;
   33,   4266,   20184,   20184,    4266,      33;
   65,  24548,  210860,  376768,  210860,   24548,     65;
  129, 143814, 2183652, 6233352, 6233352, 2183652, 143814, 129;
		

Links

• G. C. Greubel, Rows n = 0..40 of triangle, flattened
• A. Lakhtakia, R. Messier, V. K. Varadan, V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Programs

Mathematica

r = 2; q = 1; p[x_, n_] = 2^n*(1-x)^(n+1)*LerchPhi[x, -n, 1/2];
b:= Table[CoefficientList[Series[p[x, n], {x, 0, 30}], x], {n, 0, 20}];
T[n_, m_]:= (r^(n-m)*q^m + r^m*q^(n-m))*b[[n+1]][[m+1]];
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, May 08 2019 *)

Formula

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).

Extensions

Edited by G. C. Greubel, May 08 2019