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%I #21 Nov 29 2023 07:51:11
%S 1,0,1,0,1,1,0,1,3,1,0,1,6,5,1,0,1,10,16,7,1,0,1,15,40,30,9,1,0,1,21,
%T 85,100,48,11,1,0,1,28,161,280,196,70,13,1,0,1,36,280,686,672,336,96,
%U 15,1,0,1,45,456,1512,2016,1344,528,126,17,1
%N Expansion of (2*x)/(sqrt((1-x)^2-4*x^2*y)+3*x-1).
%H Robert Israel, <a href="/A336541/b336541.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%F T(n,m) = C(n-1,n-m)*Sum_{k=0..n} C(n-m,m-k)*k/m, T(0,0)=1, T(0,m)=0, m>0.
%F Sum_{m=0..n} T(n,m) = A005773(n).
%F m*T(m, n) + (-2*m + 6)*T(m - 3, n) + (4*m - 12)*T(m - 3, n - 2) + (7*m - 21)*T(m - 3, n - 1) + (5*m - 9)*T(m - 2, n) + (-2*m + 9)*T(m - 2, n - 1) + (-4*m + 3)*T(m - 1, n) - m*T(m - 1, n - 1)= 0 for m>=3, n>=2. - _Robert Israel_, Oct 05 2020
%e 1,
%e 0,1,
%e 0,1, 1,
%e 0,1, 3, 1,
%e 0,1, 6, 5, 1,
%e 0,1,10,16, 7, 1,
%e 0,1,15,40, 30, 9, 1,
%e 0,1,21,85,100,48,11,1
%p T:= proc(m,n) option remember;
%p if m < n then 0
%p elif m = n then 1
%p elif n=0 then 0
%p elif n=1 then 1
%p else
%p (-(-2*m + 6)*procname(m - 3, n) - (4*m - 12)*procname(m - 3, n - 2) - (7*m - 21)*procname(m - 3, n - 1) - (5*m - 9)*procname(m - 2, n) - (-2*m + 9)*procname(m - 2, n - 1) - (-4*m + 3)*procname(m - 1, n) + m*procname(m - 1, n - 1))/m
%p fi
%p end proc:
%p seq(seq(T(n,m),m=0..n),n=0..20); # _Robert Israel_, Oct 05 2020
%t T[n_, m_] := If[n == m, 1, If[m == 0, 0, Binomial[n-1, n-m]*
%t Sum[Binomial[n-m, m-k]*k, {k, 0, n}]/m]];
%t Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2023 *)
%o (Maxima)
%o T(n,m):=if m=0 and n=0 then 1 else if m=0 then 0 else (binomial(n-1,n-m)*sum(binomial(n-m,m-k)*k,k,0,n))/m;
%Y Cf. A005773.
%K nonn,tabl
%O 0,9
%A _Vladimir Kruchinin_, Oct 04 2020
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