OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = [x^k]( p(x, n) ), where p(x, n) = (-1)^n*(1-2*x)^(n+1)*Sum_{j >= 0} (3*j+2)^n*(2*x)^j, or p(x, n) = (-2)^n * (1-2*x)^(n+1) * LerchPhi(2*x, -n, 2/3).
Sum_{k=0..n} T(n, k) = A151919(n) (row sums).
EXAMPLE
Triangle begins as:
1;
-2, -2;
4, 26, 4;
-8, -186, -240, -8;
16, 1090, 4524, 2008, 16;
-32, -5866, -57992, -85424, -16288, -32;
64, 30354, 616452, 2099504, 1423968, 130848, 64;
-128, -154202, -5902944, -39122296, -61925632, -22159968, -1048064, -128;
MATHEMATICA
m=12; p[x_, n_]= (-1)^n*(1-2*x)^(n+1)*Sum[(3*j+2)^n*(2*x)^j, {j, 0, m+2}];
T[n_, k_]:= Coefficient[Series[p[x, n], {x, 0, n}], x, k];
Table[T[n, k], {n, 0, m}, {k, 0, n}]//Flatten
PROG
(Magma)
m:=12;
R<x>:=PowerSeriesRing(Integers(), m+2);
p:= func< n, x | (-1)^n*(1-2*x)^(n+1)*(&+[(3*j+2)^n*(2*x)^j: j in [0..n]]) >;
T:= func< n, k | Coefficient(R!( p(n, x) ), k) >;
[T(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, May 26 2024
(SageMath)
m=12
def p(x, n): return (-1)^n*(1-2*x)^(n+1)*sum((3*j+2)^n*(2*x)^j for j in range(n+1))
def T(n, k): return ( p(x, n) ).series(x, n+1).list()[k]
flatten([[T(n, k) for k in range(n+1)] for n in range(m+1)]) # G. C. Greubel, May 26 2024
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Jan 12 2009
EXTENSIONS
Edited by G. C. Greubel, May 26 2024
STATUS
approved