A138076 Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

1, -1, 1, 1, -6, 1, -1, 23, -23, 1, 1, -76, 230, -76, 1, -1, 237, -1682, 1682, -237, 1, 1, -722, 10543, -23548, 10543, -722, 1, -1, 2179, -60657, 259723, -259723, 60657, -2179, 1, 1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1, -1, 19673, -1756340, 21707972, -69413294, 69413294, -21707972, 1756340, -19673, 1

Offset: 0

Author

Roger L. Bagula, Nov 26 2009

Keywords

• sign
• tabl

Comments

Former name: A signed version of A060187 obtained by taking the Z-transform of p(t,x) = exp(t*(1+2*x)). - G. C. Greubel, Jul 21 2024

Examples

			Triangle begins as:
   1;
  -1,     1;
   1,    -6,      1;
  -1,    23,    -23,        1;
   1,   -76,    230,      -76,       1;
  -1,   237,  -1682,     1682,    -237,        1;
   1,  -722,  10543,   -23548,   10543,     -722,      1;
  -1,  2179, -60657,   259723, -259723,    60657,  -2179,     1;
   1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1;
		

Links

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

Crossrefs

Cf. A000165, A002436, A060187, A177043, A178118.

Programs

Magma

A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A138076:= func< n,k | (-1)^(n+k)*A060187(n+1,k+1) >;
[A138076(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2024

Mathematica

p[t_] = Exp[t]*x/(Exp[2*t] + x);
Table[CoefficientList[(n!*(1+x)^(n+1)/x)*SeriesCoefficient[Series[p[ t], {t,0,30}], n], x], {n,0,12}]//Flatten

SageMath

@CachedFunction
def t(n,k): # t = A060187
    if k==1 or k==n: return 1
    return (2*(n-k)+1)*t(n-1, k-1) + (2*k-1)*t(n-1, k)
def A138076(n,k): return (-1)^(n+k)*t(n+1,k+1)
flatten([[A138076(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2024

Formula

T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

From G. C. Greubel, Jul 21 2024: (Start)

T(2*n, n) = (-1)^n * A177043(n).

Sum_{k=0..n} T(n, k) = (1/2)*(1 + (-1)^n)*(-1)^floor((n+ 1)/2) * A002436(floor(n/2)).

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A000165(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A178118(n+1). (End)