A225434 Apply the triangle-to-triangle transformation described in the Comments in A159041 to the triangle in A142459.
1, 1, 1, 1, -58, 1, 1, -307, -307, 1, 1, -1556, 12006, -1556, 1, 1, -7805, 140722, 140722, -7805, 1, 1, -39054, 1461615, -5647300, 1461615, -39054, 1, 1, -195303, 14287093, -109642851, -109642851, 14287093, -195303, 1, 1, -976552, 135028828, -1838120344, 4873361350, -1838120344, 135028828, -976552, 1
Offset: 0
Examples
The triangle begins: 1; 1, 1; 1, -58, 1; 1, -307, -307, 1; 1, -1556, 12006, -1556, 1; 1, -7805, 140722, 140722, -7805, 1; 1, -39054, 1461615, -5647300, 1461615, -39054, 1; 1, -195303, 14287093, -109642851, -109642851, 14287093, -195303, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Maple
See A159041.
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Mathematica
(* First program *) t[n_, k_, m_]:= t[n, k, m]= If[k==0 || k==n, 1, (m*(n+1)-m*(k+1)+1)*t[n-1,k-1,m] + (m*(k+1)-(m-1))*t[n-1,k,m] ]; (* t(n,k,4)=A142459 *) 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,4], (-1)^(n-i+1)*t[n,i,4]]], {i,0,n}]/(1-x); Flatten[Table[CoefficientList[p[x, n], x], {n,0,12}]] (* Second program *) t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*(n+1)-m*(k+1)+1)*t[n-1,k-1,m] + (m*(k+1)-(m-1))*t[n-1,k,m]]; T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] + (-1)^k*t[n+2,k+1,4], T[n, n-k]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 19 2022 *)
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Sage
@CachedFunction def T(n, k, m): if (k==1 or k==n): return 1 else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m) def A142459(n,k): return T(n,k,4) @CachedFunction def A225434(n,k): if (k==0 or k==n): return 1 elif (k <= (n//2)): return A225434(n,k-1) + (-1)^k*A142459(n+2,k+1) else: return A225434(n,n-k) flatten([[A225434(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 19 2022
Formula
A triangle of polynomial coefficients: p(x,n) = Sum_{i=0..n} ( x^i * if(i = floor(n/2) and (n mod 2) = 0, 0, if(i <= floor(n/2), (-1)^i*A142459(n+1, i+1), (-1)^(n-i+1)*A142459(n+1, i+1) ) )/(1-x).
T(n, k) = T(n,k-1) + (-1)^k*A142459(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - G. C. Greubel, Mar 19 2022
Extensions
Edited by N. J. A. Sloane, May 11 2013