A395149 Triangle of coefficients of characteristic polynomial of negative Pascal matrix with (i+1,j+1)-th entry -C(i+j+2,j).

1, 1, 1, 1, 5, 1, 1, 20, 20, 1, 1, 76, 275, 76, 1, 1, 286, 3431, 3431, 286, 1, 1, 1078, 42238, 131714, 42238, 1078, 1, 1, 4081, 527136, 4893890, 4893890, 527136, 4081, 1, 1, 15521, 6715171, 185250188, 527873698, 185250188, 6715171, 15521, 1

Offset

0,5

Comments

Row sums are A005130(n+1). See Theorems 32 and 34 in Krattenthaler reference.

Conjecture: The alternating row sum of index 2n is A266091(n)^4 / 3^(2*n) / (2*n+1). For example, 1 - 76 + 275 - 76 + 1 = 15^4 / 3^4 / 5.

Links

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

Christian Krattenthaler, Advanced Determinant Calculus, Sém. Loth. Comb. 42.

Example

The first few rows are
  1;
  1, 1;
  1, 5, 1;
  1, 20, 20, 1;
  1, 76, 275, 76, 1;
  1, 286, 3431, 3431, 286, 1;
  ...

Prog

(SageMath)
def m(d):
    return matrix(d, d, lambda i, j: -binomial(i + j + 2, j))
[list(m(d).charpoly()) for d in range(12)]

Crossrefs

Similar to A045912 and A395150.

Sequence in context: A383753 A144397 A047909 * A171243 A111577 A176242

Adjacent sequences: A395146 A395147 A395148 * A395150 A395151 A395153

Keyword

nonn,tabl

Author

F. Chapoton, Apr 14 2026

Status

approved