A101980 - OEIS

%I #9 Jul 19 2024 19:04:44

%S 0,1,0,-1,4,0,4,-9,9,0,-33,64,-36,16,0,456,-825,400,-100,25,0,-9460,

%T 16416,-7425,1600,-225,36,0,274800,-463540,201096,-40425,4900,-441,49,

%U 0,-10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0,530052880,-862143345,356140800,-66749760,7239456

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

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

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

%e Rows begin:

%e [0],

%e [1,0],

%e [ -1,4,0],

%e [4,-9,9,0],

%e [ -33,64,-36,16,0],

%e [456,-825,400,-100,25,0],

%e [ -9460,16416,-7425,1600,-225,36,0],

%e [274800,-463540,201096,-40425,4900,-441,49,0],

%e [ -10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0],...

%e and equal the term-by-term product of column 0:

%e A101981 = {0,1,-1,4,-33,456,-9460,274800,-10643745,...}

%e with the rows of the squared Pascal's triangle (A008459):

%e [0],

%e [1*1^2, 0*1^2],

%e [ -1*1^2, 1*2^2, 0*1^2],

%e [4*1^2, -1*3^2, 1*3^2, 0*1^2],

%e [ -33*1^2, 4*4^2, -1*6^2, 1*4^2, 0*1^2],

%e [456*1^2, -33*5^2, 4*10^2, -1*10^2, 1*5^2, 0*1^2],...

%o (PARI) {T(n,k)=if(n<k||k<0,0,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,k+1]))}

%Y Cf. A008459, A002190, A101981, A101982.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Dec 23 2004