OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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.
EXAMPLE
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;
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 *)
PROG
(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
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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, May 07 2013
EXTENSIONS
Edited by N. J. A. Sloane, May 11 2013
Edited by G. C. Greubel, Mar 18 2022
STATUS
approved