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

0,5

Links

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

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

Example

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;

Maple

See A159041.

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 *)

Prog

(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

Crossrefs

Cf. A142459, A159041.

Sequence in context: A065869 A301558 A022082 * A225415 A033378 A259082

Adjacent sequences: A225431 A225432 A225433 * A225435 A225436 A225437

Keyword

sign,tabl

Author

Roger L. Bagula, May 07 2013

Extensions

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

Status

approved