A336727 - OEIS

%I #28 Aug 08 2020 01:36:24

%S 1,1,1,1,1,1,1,1,0,1,1,1,-1,-1,1,1,1,-2,-1,0,1,1,1,-3,1,5,2,1,1,1,-4,

%T 5,10,-3,0,1,1,1,-5,11,9,-38,-21,-5,1,1,1,-6,19,-4,-103,28,51,0,1,1,1,

%U -7,29,-35,-174,357,289,41,14,1,1,1,-8,41,-90,-203,1176,-131,-1262,-391,0,1

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

%H Seiichi Manyama, <a href="/A336727/b336727.txt">Antidiagonals n = 0..139, flattened</a>

%F G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).

%F A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).

%F T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).

%F (n+1) * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-2) * T(n-2,k) for n>1. - _Seiichi Manyama_, Aug 08 2020

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

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

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

%e   1, -1,  -1,   1,    5,   11,   19, ...

%e   1,  0,   5,  10,    9,   -4,  -35, ...

%e   1,  2,  -3, -38, -103, -174, -203, ...

%e   1,  0, -21,  28,  357, 1176, 2575, ...

%t T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j],(-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* _Amiram Eldar_, Aug 02 2020 *)

%o (PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}

%o (PARI) {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}

%o (PARI) {T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

%Y Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.

%Y Main diagonal gives A336728.

%Y Cf. A243631, A307884, A336708, A336709.

%K sign,tabl

%O 0,18

%A _Seiichi Manyama_, Aug 02 2020