A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

1, 1, 58, 1, 1, 1556, 12006, 1556, 1, 1, 39054, 1461615, 5647300, 1461615, 39054, 1, 1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1, 1, 24414050, 11462721645, 414730580760, 3221733789330, 6783391017228, 3221733789330, 414730580760, 11462721645, 24414050, 1

Offset: 1

Author

Roger L. Bagula, May 07 2013

Keywords

• nonn
• tabf
• easy

Examples

			Triangle begins:
  1;
  1,     58,         1;
  1,   1556,     12006,       1556,          1;
  1,  39054,   1461615,    5647300,    1461615,      39054,         1;
  1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Crossrefs

The m=4 triangle in the sequence A034870 (m=0), A171692 (m=1), A225076 (m=2), A225398 (m=3).

Programs

Mathematica

(* First program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1,(m*n-m*k+1)*t[n-1, k-1, m] + (m*k-(m-1))*t[n-1, k, m]];
T[n_, k_]:= T[n, k] = t[n+1, k+1,4]; (* t(n,k,4) = A142459 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n,1,14,2}]]
(* Second program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k-m+1)*t[n-1,k,m]]; (* t(n,k,4) = A142459 *)
T[n_, k_]:= T[n, k]= Sum[ (-1)^(k-j-1)*t[2*n,j+1,4], {j,0,k-1}];
Table[T[n, k], {n,12}, {k,2*n-1}]//Flatten (* G. C. Greubel, Mar 19 2022 *)

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)
def A225415(n,k): return sum( (-1)^(k-j-1)*A142459(2*n, j+1) for j in (0..k-1) )
flatten([[A225415(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

Formula

T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142459(2*n, j+1). - G. C. Greubel, Mar 19 2022

Extensions

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