A361290 - OEIS

%I #45 Mar 26 2023 11:14:36

%S 0,0,1,0,1,0,0,1,2,0,0,1,4,4,0,0,1,6,14,8,0,0,1,8,30,48,16,0,0,1,10,

%T 52,144,164,32,0,0,1,12,80,320,684,560,64,0,0,1,14,114,600,1936,3240,

%U 1912,128,0,0,1,16,154,1008,4400,11648,15336,6528,256,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

%F T(0,k) = 0, T(1,k) = 1; 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 * sqrt(k)) for k > 0.

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

%F E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

%e Square array begins:

%e   0,  0,   0,   0,    0,    0, ...

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

%e   0,  2,   4,   6,    8,   10, ...

%e   0,  4,  14,  30,   52,   80, ...

%e   0,  8,  48, 144,  320,  600, ...

%e   0, 16, 164, 684, 1936, 4400, ...

%o (PARI) T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

%Y Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.

%Y Main diagonal gives A360766.

%Y Cf. A361432.

%K nonn,easy,tabl

%O 0,9

%A _Seiichi Manyama_, Mar 11 2023