A101981 Column 0 of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).

0, 1, -1, 4, -33, 456, -9460, 274800, -10643745, 530052880, -32995478376, 2510382661920, -229195817258100, 24730000147369440, -3113066087894608560, 452168671458789789504, -75059305956331837485345, 14121026957032156557396000, -2988687741694684876495689040

Offset: 0

Author

Paul D. Hanna, Dec 23 2004

Keywords

• sign

Comments

This sequence is a signed version of A002190 and is related to Bessel functions.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• O. Arizmendi, T. Hasebe, F. Lehner, C. Vargas, Relations between cumulants in noncommutative probability, arXiv preprint arXiv:1408.2977 [math.CO], 2014.
• Christian Günther, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.

Crossrefs

Cf. A008459, A002190, A101980, A101982.

Programs

Maple

a:= n-> (-1)^(n+1)*coeff (series (-ln(BesselJ(0, 2*sqrt(x))), x, n+1), x, n)*(n!)^2: seq (a(n), n=0..30); # Alois P. Heinz, Oct 27 2012

Mathematica

a[n_] := (-1)^(n+1) n!^2 SeriesCoefficient[-Log[BesselJ[0, 2 Sqrt[x]]], {x, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 26 2018 *)

PARI

{a(n)=sum(m=1,n,(-1)^(m-1)* (matrix(n+1,n+1,i,j,if(i>j,binomial(i-1,j-1)^2))^m/m)[n+1,1])}

Formula

a(n) = (-1)^(n+1)*A002190(n) for n>=0.

a(n) = 1 - Sum_{j=1..k-1} binomial(k, j)*binomial(k-1, j-1)*a(j) for n >= 1. See Günther & Schmidt link p.5. - Michel Marcus, Jun 17 2017