cp's OEIS Frontend

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

%I A101981 #20 Jul 26 2018 06:35:26
%S A101981 0,1,-1,4,-33,456,-9460,274800,-10643745,530052880,-32995478376,
%T A101981 2510382661920,-229195817258100,24730000147369440,
%U A101981 -3113066087894608560,452168671458789789504,-75059305956331837485345,14121026957032156557396000,-2988687741694684876495689040
%N A101981 Column 0 of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).
%C A101981 This sequence is a signed version of A002190 and is related to Bessel functions.
%H A101981 Alois P. Heinz, <a href="/A101981/b101981.txt">Table of n, a(n) for n = 0..100</a>
%H A101981 O. Arizmendi, T. Hasebe, F. Lehner, C. Vargas, <a href="http://arxiv.org/abs/1408.2977">Relations between cumulants in noncommutative probability</a>, arXiv preprint arXiv:1408.2977 [math.CO], 2014.
%H A101981 Christian Günther, Kai-Uwe Schmidt, <a href="http://arxiv.org/abs/1602.01750">Lq norms of Fekete and related polynomials</a>, arXiv:1602.01750 [math.NT], 2016.
%F A101981 a(n) = (-1)^(n+1)*A002190(n) for n>=0.
%F A101981 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
%p A101981 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
%t A101981 a[n_] := (-1)^(n+1) n!^2 SeriesCoefficient[-Log[BesselJ[0, 2 Sqrt[x]]], {x, 0, n}];
%t A101981 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jul 26 2018 *)
%o A101981 (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])}
%Y A101981 Cf. A008459, A002190, A101980, A101982.
%K A101981 sign
%O A101981 0,4
%A A101981 _Paul D. Hanna_, Dec 23 2004