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%I #17 Nov 15 2020 13:00:41
%S 1,2,2,3,14,3,4,44,44,4,5,100,238,100,5,6,190,828,828,190,6,7,322,
%T 2233,4092,2233,322,7,8,504,5096,14872,14872,5096,504,8,9,744,10332,
%U 43992,70070,43992,10332,744,9,10,1050,19176,112200,260780,260780,112200,19176,1050,10
%N Triangle T(n,m) = (2*m*n+2*n-2*m^2+1)*C(2*n+2,2*m+1)/(4*n+2).
%F G.f.: (1/(1-x-x*y-4*x^2*y/(1-x-x*y)))^2.
%F T(n,m) = Sum_{k=0..n} C(n+1,2*k+1)*C(n-2*k,m-k)*(k+1)*4^k.
%F A045563(n) = (Sum_{m=0..n} T(n,m))/2^n.
%e 1,
%e 2, 2,
%e 3, 14, 3,
%e 4, 44, 44, 4,
%e 5, 100, 238, 100, 5,
%e 6, 190, 828, 828, 190, 6,
%e 7, 322, 2233, 4092, 2233, 322, 7
%t Table[Sum[Binomial[n + 1, 2 k + 1] Binomial[n - 2 k, m - k] (k + 1)*4^k, {k, 0, n} ], {n, 0, 9}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 04 2020 *)
%o (Maxima)
%o T(n,m):=((2*m*n+2*n-2*m^2+1)*binomial(2*n+2,2*m+1))/(4*n+2);
%Y Cf. A045623, A086645.
%Y 2nd column=2*A002412.
%K nonn,tabl
%O 0,2
%A _Vladimir Kruchinin_, Nov 01 2020
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