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%I #38 Feb 06 2021 22:33:30
%S 1,1,1,1,3,1,1,11,6,1,1,43,46,10,1,1,171,386,130,15,1,1,683,3366,1870,
%T 295,21,1,1,2731,29866,28234,6455,581,28,1,1,10923,267086,437350,
%U 149031,17941,1036,36,1
%N Square table read by upwards antidiagonals: T(m,n) = A103438(2*m-1,n)/A103438(1,n) for m>=1, n>=1.
%F Let a(i,m) = ((-2)^i)*Sum_{j=0..i} C(2*m,i-j)*C(i+j,j)*((i-j)/(i+j))*B(2*m-i+j), B(s) = A027641(s)/A027642(s) the Bernoulli numbers and N = n*(n+1)/2, then T(m,n) = (1/(2*m))*Sum_{i=2..m} a(i,m)*N^(i-1)}.
%e T(3,4) = A103438(2*3-1,4)/A103438(1,4) = 1300/10 = 130.
%e By formula: a(2,3) = 4*15*1*1*B(4) = -2 and a(3,3) = (-8)*15*4*(2/4)*B(4) = 8 yields T(3,n) = (-N+4*N^2)/3. Since N = 4*5/2 = 10, T(3,4) = (4*10^2-10)/3 = 130.
%e Table begins:
%e m\n| 1 2 3 4 5 6 7
%e ---+-----------------------------------------------------
%e 1 | 1 1 1 1 1 1 1
%e 2 | 1 3 6 10 15 21 28
%e 3 | 1 11 46 130 295 581 1036
%e 4 | 1 43 386 1870 6455 17941 42868
%e 5 | 1 171 3366 28234 149031 586341 1880956
%e 6 | 1 683 29866 437350 3546775 19809461 85475908
%e 7 | 1 2731 267086 6871138 85960967 683338501 3972825676
%Y Cf. A103438.
%K nonn,tabl
%O 1,5
%A _Franz Vrabec_, Dec 24 2020
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