cp's OEIS Frontend

A336727 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.

%I A336727 #28 Aug 08 2020 01:36:24
%S A336727 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 A336727 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 A336727 -7,29,-35,-174,357,289,41,14,1,1,1,-8,41,-90,-203,1176,-131,-1262,-391,0,1
%N A336727 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 A336727 Seiichi Manyama, <a href="/A336727/b336727.txt">Antidiagonals n = 0..139, flattened</a>
%F A336727 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).
%F A336727 A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).
%F A336727 T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).
%F A336727 (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 A336727   1,  1,   1,   1,    1,    1,    1, ...
%e A336727   1,  1,   1,   1,    1,    1,    1, ...
%e A336727   1,  0,  -1,  -2,   -3,   -4,   -5, ...
%e A336727   1, -1,  -1,   1,    5,   11,   19, ...
%e A336727   1,  0,   5,  10,    9,   -4,  -35, ...
%e A336727   1,  2,  -3, -38, -103, -174, -203, ...
%e A336727   1,  0, -21,  28,  357, 1176, 2575, ...
%t A336727 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 A336727 (PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}
%o A336727 (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 A336727 (PARI) {T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}
%Y A336727 Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.
%Y A336727 Main diagonal gives A336728.
%Y A336727 Cf. A243631, A307884, A336708, A336709.
%K A336727 sign,tabl
%O A336727 0,18
%A A336727 _Seiichi Manyama_, Aug 02 2020