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

1,3

Links

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

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)

Example

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;

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 *)

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 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

Crossrefs

Cf. A034870, A171692, A225076.

Sequence in context: A351221 A154229 A225433 * A037936 A277642 A160147

Adjacent sequences: A225395 A225396 A225397 * A225399 A225400 A225401

Keyword

nonn,tabf

Author

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

Extensions

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

Status

approved