A154594 Triangle read by rows: 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.

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, 256, 776642, 53083228, 619239464, 1884138544, 1615232096, 331200832, 8387456, 256

Offset: 0

Author

Roger L. Bagula, Jan 12 2009

Keywords

• sign
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A151919 (row sums), A154593.

Programs

Magma

m:=12;
R:=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

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

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

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).

Extensions

Edited by G. C. Greubel, May 26 2024