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%I #10 Feb 02 2022 23:39:33
%S 0,2,0,4,4,0,6,20,6,0,8,56,56,8,0,10,120,252,120,10,0,12,220,792,792,
%T 220,12,0,14,364,2002,3432,2002,364,14,0,16,560,4368,11440,11440,4368,
%U 560,16,0,18,816,8568,31824,48620,31824,8568,816,18,0,20,1140,15504
%N Triangle T(n, k) read by rows: binomial(2n, 2k+1).
%C A subset of Pascal's triangle A007318 with only even elements.
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
%F From _Peter Bala_, Jan 31 2022: (Start)
%F T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2), with T(0,0) = 0, T(1,0) = 2 and T(n,k) = 0 if k < 0 or if k > n-1.
%F n-th row polynomial R(n,x) = (1/(2*sqrt(x)))*( (1 + sqrt(x))^(2*n) - (1 - sqrt(x))^(2*n) ).
%F O.g.f.: A(x,t) = 2*t/(1 - 2*(x + 1)*t + (x - 1)^2*t^2) = 2*t + (4 + 4*x)*t^2 + (6 + 20*x + 6*x^2)*t^3 + ....
%F G.f.: (1/sqrt(x))*sinh(t)*sinh(sqrt(x)*t) = 2*t^2/2! + (4 + 4*x)*t^4/4! + (6 + 20*x^2 + 6*x^3)*t^6/6! + ....
%F O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k+1)*x^k )/(1 - x)^(2*n) = 1/(2*sqrt(x))*((1 - sqrt(x))^(-2*n) - (1 + sqrt(x))^(-2*n)).
%F With a different offset, 2/(x-4)*A(x/(x-4), t*(x-4)/4) = t/(1 + t*(2 - x) + t^2) is a g.f. of A053122.
%F Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 1:
%F (1/2)*(N^2 + N)^(2*n) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n-1)*S(4*n-1,N). Some examples are given below. (End)
%e Triangle begins
%e 0;
%e 2, 0;
%e 4, 4, 0;
%e 6, 20, 6, 0;
%e 8, 56, 56, 8, 0;
%e 10, 120, 252, 120, 10, 0;
%e 12, 220, 792, 792, 220, 12, 0;
%e 14, 364, 2002, 3432, 2002, 364, 14, 0;
%e 16, 560, 4368, 11440, 11440, 4368, 560, 16, 0;
%e ...
%e From _Peter Bala_, Jan 30 2022: (Start)
%e (1/2)*(N^2 + N)^2 = 2*Sum_{j = 1..N} j^3.
%e (1/2)*(N^2 + N)^4 = 4*Sum_{j = 1..N} j^5 + 4*Sum_{j = 1..N} j^7.
%e (1/2)*(N^2 + N)^6 = 6*Sum_{j = 1..N} j^7 + 20*Sum_{j = 1..N} j^9 + 6*Sum_{j = 1..N} j^11.
%e (1/2)*(N^2 + N)^8 = 8*Sum_{j = 1..N} j^9 + 56*Sum_{j = 1..N} j^11 + 56*Sum_{j = 1..N} j^13 + 8*Sum_{j = 1..N} j^15. (End)
%Y Cf. A091042, A086645, A103327.
%K nonn,tabl
%O 0,2
%A _Ralf Stephan_, Feb 06 2005
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