|
|
A165890
|
|
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1) * Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2, read by rows.
|
|
4
|
|
|
1, 1, -2, 1, 1, 0, -2, 0, 1, 1, 10, 15, -52, 15, 10, 1, 1, 44, 484, -44, -970, -44, 484, 44, 1, 1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1, 1, 472, 58586, 682040, 2085135, -682512, -4287444, -682512, 2085135, 682040, 58586, 472, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1)*Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2.
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (2^(n-1)*(1-x)^(n+2)*LerchPhi(x, -n+1, 1/2))^2.
Sum_{k=0..n} T(n, k) = 0^n.
|
|
EXAMPLE
|
Irregular triangle begins as:
1;
1, -2, 1;
1, 0, -2, 0, 1;
1, 10, 15, -52, 15, 10, 1;
1, 44, 484, -44, -970, -44, 484, 44, 1;
1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1;
|
|
MATHEMATICA
|
p[n_, x_]:= p[n, x]= If[n==0, 1, (2^(n-1)*(1-x)^(n+1)*LerchPhi[x, -n+1, 1/2])^2];
Table[CoefficientList[p[n, x], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)
|
|
PROG
|
(Sage)
def p(n, x): return (1-x)^(2*n+2)*sum( (2*j+1)^(n-1)*x^j for j in (0..2*n+2) )^2
def T(n, k): return ( p(n, x) ).series(x, 2*n+2).list()[k]
flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|