A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, -22, 1, 1, -75, -75, 1, 1, -236, 1446, -236, 1, 1, -721, 9822, 9822, -721, 1, 1, -2178, 58479, -201244, 58479, -2178, 1, 1, -6551, 325061, -2160227, -2160227, 325061, -6551, 1, 1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1

Offset: 0

Author

Roger L. Bagula, May 07 2013

Keywords

• sign
• tabl

Examples

			The triangle begins:
  1;
  1,      1;
  1,    -22,       1;
  1,    -75,     -75,         1;
  1,   -236,    1446,      -236,        1;
  1,   -721,    9822,      9822,     -721,         1;
  1,  -2178,   58479,   -201244,    58479,     -2178,       1;
  1,  -6551,  325061,  -2160227, -2160227,    325061,   -6551,      1;
  1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1;
		

Links

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

Crossrefs

Cf A007318, A060187, A159041.

Programs

Mathematica

(* First program *)
q[x_, n_]= (1-x)^(n+1)*Sum[(2*m+1)^n*x^m, {m, 0, Infinity}];
t[n_, m_]:= t[n, m]= Table[CoefficientList[q[x, k], x], {k,0,15}][[n+1, m+1]];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i <= Floor[n/2], (-1)^i*t[n, i], (-1)^(n-i+1)*t[n, i]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,10}]]
(* Second Program *)
A060187[n_, k_]:= Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i,k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] +(-1)^k*A060187[n+2, k+1], T[n, n-k] ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

Sage

def A060187(n,k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
@CachedFunction
def A225356(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225356(n,k-1) + (-1)^k*A060187(n+2,k+1)
    else: return A225356(n,n-k)
flatten([[A225356(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

Extensions

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 18 2022