A045912 Triangle of coefficients of characteristic polynomial of negative Pascal matrix with (i,j)-th entry -C(i+j-2,i-1).

1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 29, 72, 29, 1, 1, 99, 626, 626, 99, 1, 1, 351, 6084, 13869, 6084, 351, 1, 1, 1275, 64974, 347020, 347020, 64974, 1275, 1, 1, 4707, 744193, 9952274, 21537270, 9952274, 744193, 4707, 1, 1, 17577, 8965323, 321541977, 1545936516, 1545936516, 321541977, 8965323, 17577, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.

W. F. Lunnon, The Pascal matrix, Fib. Quart. vol. 15 (1977) pp. 201-204.

Luca Guido Molinari, Graphene nanocones and Pascal matrices, arXiv:2206.14428 [math.CO], 2022.

Example

1;
1,1;
1,3,1;
1,9,9,1;
1,29,72,29,1;
...

Mathematica

P[n_] := Table[Binomial[i + j - 2, i - 1], {i, 1, n}, {j, 1, n}];
row[0] = {1};
row[n_] := CoefficientList[ CharacteristicPolynomial[P[n], x], x] // Abs;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 09 2018 *)

Prog

(PARI) T(n, k)=if(n<0, 0, (-1)^(n+k)*polcoeff(charpoly(matrix(n, n, i, j, binomial(i+j-2, i-1))), k))
(PARI) T(n, k)=if(n<0, 0, polcoeff(charpoly(-matrix(n, n, i, j, binomial(i+j-2, i-1))), k))

Crossrefs

Sum of k-th row is A006366(n). Columns give A006134, A006135, A006136.

Sequence in context: A144493 A118180 A176482 * A290554 A267264 A267197

Adjacent sequences: A045909 A045910 A045911 * A045913 A045914 A045915

Keyword

nonn,tabl

Author

Fred Lunnon, Dec 11 1999

Status

approved