A395149 - OEIS

%I #20 Apr 27 2026 00:28:42

%S 1,1,1,1,5,1,1,20,20,1,1,76,275,76,1,1,286,3431,3431,286,1,1,1078,

%T 42238,131714,42238,1078,1,1,4081,527136,4893890,4893890,527136,4081,

%U 1,1,15521,6715171,185250188,527873698,185250188,6715171,15521,1

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

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

%C 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.

%H Christian Krattenthaler, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s42kratt.html">Advanced Determinant Calculus</a>, Sém. Loth. Comb. 42.

%e The first few rows are

%e   1;

%e   1, 1;

%e   1, 5, 1;

%e   1, 20, 20, 1;

%e   1, 76, 275, 76, 1;

%e   1, 286, 3431, 3431, 286, 1;

%e   ...

%o (SageMath)

%o def m(d):

%o     return matrix(d, d, lambda i, j: -binomial(i + j + 2, j))

%o [list(m(d).charpoly()) for d in range(12)]

%Y Similar to A045912 and A395150.

%K nonn,tabl

%O 0,5

%A _F. Chapoton_, Apr 14 2026