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

%I A101980 #9 Jul 19 2024 19:04:44
%S A101980 0,1,0,-1,4,0,4,-9,9,0,-33,64,-36,16,0,456,-825,400,-100,25,0,-9460,
%T A101980 16416,-7425,1600,-225,36,0,274800,-463540,201096,-40425,4900,-441,49,
%U A101980 0,-10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0,530052880,-862143345,356140800,-66749760,7239456
%N A101980 Matrix logarithm of A008459 (squared entries of Pascal's triangle), read by rows.
%C A101980 Column 0 (A101981) is essentially a signed offset version of A002190 and is related to Bessel functions. Row sums form A101982.
%F A101980 T(n, k) = A101981(n-k)*C(n, k)^2.
%e A101980 Rows begin:
%e A101980 [0],
%e A101980 [1,0],
%e A101980 [ -1,4,0],
%e A101980 [4,-9,9,0],
%e A101980 [ -33,64,-36,16,0],
%e A101980 [456,-825,400,-100,25,0],
%e A101980 [ -9460,16416,-7425,1600,-225,36,0],
%e A101980 [274800,-463540,201096,-40425,4900,-441,49,0],
%e A101980 [ -10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0],...
%e A101980 and equal the term-by-term product of column 0:
%e A101980 A101981 = {0,1,-1,4,-33,456,-9460,274800,-10643745,...}
%e A101980 with the rows of the squared Pascal's triangle (A008459):
%e A101980 [0],
%e A101980 [1*1^2, 0*1^2],
%e A101980 [ -1*1^2, 1*2^2, 0*1^2],
%e A101980 [4*1^2, -1*3^2, 1*3^2, 0*1^2],
%e A101980 [ -33*1^2, 4*4^2, -1*6^2, 1*4^2, 0*1^2],
%e A101980 [456*1^2, -33*5^2, 4*10^2, -1*10^2, 1*5^2, 0*1^2],...
%o A101980 (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 A101980 Cf. A008459, A002190, A101981, A101982.
%K A101980 sign,tabl
%O A101980 0,5
%A A101980 _Paul D. Hanna_, Dec 23 2004