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

1,3

Links

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

Formula

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

Example

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;

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 *)

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)
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

Crossrefs

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

Sequence in context: A301558 A022082 A225434 * A033378 A259082 A366457

Adjacent sequences: A225412 A225413 A225414 * A225416 A225417 A225418

Keyword

nonn,tabf

Author

Roger L. Bagula, May 07 2013

Extensions

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

Status

approved