A225532 Triangle T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)), read by rows.
1, 1, 1, 1, 26, 1, 1, 27, 27, 1, 1, 120, 1192, 120, 1, 1, 121, 1312, 1312, 121, 1, 1, 502, 14609, 88736, 14609, 502, 1, 1, 503, 15111, 103345, 103345, 15111, 503, 1, 1, 2036, 152638, 2205524, 9890752, 2205524, 152638, 2036, 1, 1, 2037, 154674, 2358162, 12096276, 12096276, 2358162, 154674, 2037, 1
Offset: 0
Examples
Triangle begins: 1; 1, 1; 1, 26, 1; 1, 27, 27, 1; 1, 120, 1192, 120, 1; 1, 121, 1312, 1312, 121, 1; 1, 502, 14609, 88736, 14609, 502, 1; 1, 503, 15111, 103345, 103345, 15111, 503, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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); q[n_, x_]= If[Mod[n,2]==0, (1-x)*p[n/2,x], p[(n+1)/2,x]]; Table[Abs[CoefficientList[q[(4*n +(-1)^n +5)/2, x], x]], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 29 2022 *) (* 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 *) A225483[n_, k_]:= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}]; T[n_, k_]:= If[Mod[n,2]==0, A225483[n/2, k], A225483[(n-1)/2, k] - A225483[(n - 1)/2, k-1] ]//Abs; Table[T[n, k], {n,0,10}, {k,0,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) ) @CachedFunction def A225532(n,k): if (n%2==0): return abs(A225483(n/2, k)) else: return abs( A225483((n-1)/2, k) - A225483((n-1)/2, k-1) ) flatten([[A225532(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2022
Formula
From G. C. Greubel, Mar 29 2022: (Start)
T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)).
T(n, n-k) = T(n, k). (End)
Extensions
Edited by G. C. Greubel, Mar 29 2022