A361432 - OEIS

%I #29 Mar 12 2023 08:47:53

%S 1,1,0,1,1,0,1,2,2,0,1,3,6,4,0,1,4,12,20,8,0,1,5,20,54,68,16,0,1,6,30,

%T 112,252,232,32,0,1,7,42,200,656,1188,792,64,0,1,8,56,324,1400,3904,

%U 5616,2704,128,0,1,9,72,490,2628,10000,23360,26568,9232,256,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

%H Seiichi Manyama, <a href="/A361432/b361432.txt">Antidiagonals n = 0..139, flattened</a>

%F T(0,k) = 1, T(1,k) = k; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).

%F T(n,k) = ((k + sqrt(k))^n + (k - sqrt(k))^n)/2.

%F G.f. of column k: (1 - k * x)/(1 - 2 * k * x + (k-1) * k * x^2).

%F E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).

%e Square array begins:

%e   1,  1,   1,    1,    1,     1, ...

%e   0,  1,   2,    3,    4,     5, ...

%e   0,  2,   6,   12,   20,    30, ...

%e   0,  4,  20,   54,  112,   200, ...

%e   0,  8,  68,  252,  656,  1400, ...

%e   0, 16, 232, 1188, 3904, 10000, ...

%o (PARI) T(n,k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));

%o (PARI) T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;

%Y Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.

%Y Main diagonal gives A084062.

%Y Cf. A084061, A084097.

%K nonn,tabl,easy

%O 0,8

%A _Seiichi Manyama_, Mar 11 2023