A101982 -id:A101982 - cp's OEIS frontend

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

Paul D. Hanna, Dec 23 2004

sign

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

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.

Cf. A008459, A002190, A101980, A101982.

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

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 *)

{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])}

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

A101980 Matrix logarithm of A008459 (squared entries of Pascal's triangle), read by rows.

Original entry on oeis.org

0, 1, 0, -1, 4, 0, 4, -9, 9, 0, -33, 64, -36, 16, 0, 456, -825, 400, -100, 25, 0, -9460, 16416, -7425, 1600, -225, 36, 0, 274800, -463540, 201096, -40425, 4900, -441, 49, 0, -10643745, 17587200, -7416640, 1430016, -161700, 12544, -784, 64, 0, 530052880, -862143345, 356140800, -66749760, 7239456

Offset: 0

Paul D. Hanna, Dec 23 2004

Column 0 (A101981) is essentially a signed offset version of A002190 and is related to Bessel functions. Row sums form A101982.

			Rows begin:
[0],
[1,0],
[ -1,4,0],
[4,-9,9,0],
[ -33,64,-36,16,0],
[456,-825,400,-100,25,0],
[ -9460,16416,-7425,1600,-225,36,0],
[274800,-463540,201096,-40425,4900,-441,49,0],
[ -10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0],...
and equal the term-by-term product of column 0:
A101981 = {0,1,-1,4,-33,456,-9460,274800,-10643745,...}
with the rows of the squared Pascal's triangle (A008459):
[0],
[1*1^2, 0*1^2],
[ -1*1^2, 1*2^2, 0*1^2],
[4*1^2, -1*3^2, 1*3^2, 0*1^2],
[ -33*1^2, 4*4^2, -1*6^2, 1*4^2, 0*1^2],
[456*1^2, -33*5^2, 4*10^2, -1*10^2, 1*5^2, 0*1^2],...

Cf. A008459, A002190, A101981, A101982.

{T(n,k)=if(nj,binomial(i-1,j-1)^2))^m/m)[n+1,k+1]))}

T(n, k) = A101981(n-k)*C(n, k)^2.

cp's OEIS Frontend

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

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