A395065 Triangle read by rows: coefficients of (1+(2*n-1)*x)*(1+(n-1)*x)^(n-1).

1, 1, 1, 1, 4, 3, 1, 9, 24, 20, 1, 16, 90, 216, 189, 1, 25, 240, 1120, 2560, 2304, 1, 36, 525, 4000, 16875, 37500, 34375, 1, 49, 1008, 11340, 75600, 299376, 653184, 606528, 1, 64, 1764, 27440, 264110, 1613472, 6117748, 13176688, 12353145

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Example

The first few rows are:
  1;
  1,  1;
  1,  4,  3;
  1,  9, 24,  20;
  1, 16, 90, 216, 189;
  ...

Mathematica

T[n_]:=CoefficientList[ (1 + (2 n - 1) x) * (1 + (n - 1) x)^(n - 1), x]; Array[T, 9, 0]//Flatten (* James C. McMahon, May 12 2026 *)

Prog

(SageMath)
x = polygen(ZZ, 'x')
def row(n):
    if n == 0: return [1]
    return list((1+(2*n-1)*x)*(1+(n-1)*x)**(n-1))
(PARI) row(n) = Vec((1+(2*n-1)*x)*(1+(n-1)*x)^(n-1)); \\ Michel Marcus, May 09 2026

Crossrefs

Row sums are A013499.

Main diagonal gives A176043.

Columns k=0-1 give: A000012, A000290.

Similar to A395611, A071207, A243594, A395505.

Sequence in context: A039758 A157894 A395611 * A172106 A128813 A109062

Adjacent sequences: A395060 A395063 A395064 * A395066 A395067 A395068

Keyword

nonn,tabl,easy

Author

F. Chapoton, May 08 2026

Status

approved