A209577 - OEIS

%I #11 Jun 23 2018 02:33:15

%S 1,1,1,3,2,1,5,6,3,1,9,13,10,4,1,15,28,26,15,5,1,25,56,64,45,21,6,1,

%T 41,109,146,124,71,28,7,1,67,206,319,315,216,105,36,8,1,109,382,671,

%U 758,602,349,148,45,9,1,177,697,1372,1744,1576,1056,533,201,55,10

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section.

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x) = x*u(n-1,x) + v(n-1,x),

%F v(n,x) = (x+1)*u(n-1,x) + v(n-1,x) + 1,

%F where u(1,x)=1, v(1,x)=1.

%F The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - _Thomas Baruchel_, Jun 03 2018

%e First five rows:

%e   1;

%e   1,  1;

%e   3,  2,  1;

%e   5,  6,  3,  1;

%e   9, 13, 10,  4,  1;

%e First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

%t v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]   (* A209577 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]   (* A209578 *)

%Y Cf. A209578, A208510.

%K nonn,tabl

%O 1,4

%A _Clark Kimberling_, Mar 11 2012