OFFSET
0,1
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
From G. C. Greubel, Mar 19 2022: (Start)
G.f.: a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = 65/2, b = -162/2, c = 135/2.
T(n, n-k) = T(n, k). (End)
EXAMPLE
Triangle begins as:
19;
19, 19;
19, 146, 19;
19, 759, 759, 19;
19, 3154, 10374, 3154, 19;
19, 11543, 89398, 89398, 11543, 19;
19, 39210, 615669, 1394444, 615669, 39210, 19;
19, 127303, 3747297, 16267301, 16267301, 3747297, 127303, 19;
19, 401858, 21201472, 160611806, 302914330, 160611806, 21201472, 401858, 19;
MATHEMATICA
T[n_, a_, b_, c_]:= CoefficientList[Series[a*(1+x)^n + b*(1-x)^(n+2)* PolyLog[-n-1, x]/x + 2^n*c*(1-x)^(n+1)*LerchPhi[x, -n, 1/2], {x, 0, 30}], x];
Table[T[n, 65/2, -162/2, 135/2], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)
PROG
(Sage)
m=12
def LerchPhi(x, s, a): return sum( x^j/(j+a)^s for j in (0..3*m) )
def p(n, x, a, b, c): return a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2)
def T(n, k, a, b, c): return ( p(n, x, a, b, c) ).series(x, n+1).list()[k]
flatten([[T(n, k, 65/2, -162/2, 135/2) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 28 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 19 2022
STATUS
approved