A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.

1, 1, 38, 1, 1, 676, 4806, 676, 1, 1, 10914, 362895, 1346780, 362895, 10914, 1, 1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1, 1, 2796190, 1063096365, 35677598760, 267248150610, 554291429748, 267248150610, 35677598760, 1063096365, 2796190, 1

Offset: 1

Author

Roger L. Bagula, Apr 26 2013 (Entered by N. J. A. Sloane, May 06 2013)

Keywords

• nonn
• tabf
• easy

Examples

			Triangle begins:
  1;
  1,     38,        1;
  1,    676,     4806,       676,         1;
  1,  10914,   362895,   1346780,    362895,     10914,        1;
  1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;
		

Links

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

Crossrefs

Cf. A034870, A171692, A225076.

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, 3]; (* t(n,k,3) = A142458 *)
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,3) = A142458 *)
A225398[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n,j+1,3], {j,0,k-1}];
Table[A225398[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 A142458(n, k): return T(n, k, 3)
def A225398(n,k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

Formula

From G. C. Greubel, Mar 19 2022: (Start)

T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).

T(n, n-k) = T(n, k). (End)

Extensions

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