A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.

1, 1, -26, 1, 1, -120, 1192, -120, 1, 1, -502, 14609, -88736, 14609, -502, 1, 1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1, 1, -8178, 1479727, -45541628, 424761262, -1551163136, 424761262, -45541628, 1479727, -8178, 1

Offset: 0

Author

Roger L. Bagula, May 08 2013

Keywords

• sign
• tabf

Examples

			The triangle begins:
  1;
  1,   -26,      1;
  1,  -120,   1192,     -120,       1;
  1,  -502,  14609,   -88736,   14609,     -502,      1;
  1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1;
		

Links

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

Crossrefs

Cf. A008292, A060187, A158041.

Programs

Mathematica

(* First program *)
Needs["Combinatorica`"];
p[n_, x_]:= p[n,x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1, i]*x^i, (-1)^(n-i+1)*Eulerian[n+1, i]*x^i]], {i,0,n}]/(1- x^2);
Table[CoefficientList[p[x, 2*n], x], {n,0,10}]//Flatten
(* Second program *)
A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n, k-1] + (-1)^k*A008292[n+2, k+1], f[n, n-k]]]; (* f = A159041 *)
T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}];
Table[T[n, k], {n,0,10}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 29 2022 *)

Sage

def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
@CachedFunction
def f(n, k): # A159041
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
    else: return f(n, n-k)
def A225483(n,k): return sum( (-1)^(k-j)*f(2*n+1,j) for j in (0..k) )
flatten([[A225483(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2022

Formula

T(n, k) = [x^k]( A159041(x,n)/(x+1) ).

From G. C. Greubel, Mar 29 2022: (Start)

T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j).

T(n, 2*n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 29 2022