I want to calculate the following summation for a,b;

when it is odd, 

vanishes.

MY TRY:

restart:
J:=proc(a,b,c) local u,v,p,w;
p:=(a+b+c):
if p=2*k then
u:=b+c-a: 
v:=c+a-b: 
w:=a+b-c:
return (-1)^p*sqrt(binomial(2*u,u)*binomial(2*v,v)*binomial(2*w,w)/((2*p+1)*binomial(2*p,p)))
else 
return 0
end if
end proc:
aa:=2:bb:=4:
sum(J(aa,bb,cc)*J(aa,bb,cc)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*cc+1))*psi[1,cc],cc=abs(bb-aa)..(aa+bb));

How to calculate the following for a=1,b=3

where  is the Wigner 3j symbol.

restart;
#PDE
PDE := (diff(u(x,y), x,x))-1/x*(diff(u(x, y), x)) -x^2*(diff(u(x,y), y,y))= 2;
BCs:=u(0,t) = 0, u(1,t) = 0;

ICs:=u(x,0) =0, D[2](u)(x,0)=2 ;
num_sol := pdsolve(PDE, {BCs,ICs}, numeric); 
BCs := u(0, t) = 0, u(1, t) = 0;
ICs := u(x, 0) = 0, D[2](u)(x, 0) = 2;
num_sol := pdsolve(PDE, {BCs,ICs}, numeric);

 #ODE
ODE:=diff(y(x),x,x)+16*diff(y(x),x)=0;
 BCs:=y(0) =0, y(2)=3 ;
ode_num_sol:=dsolve({ODE,BCs},numeric);

Suppose that we solved numerically an ODE or PDE like above. 
How to calculate values of function y or u at special values x like analytical solutions?

For example; u(1,1)=? y(1)=? etc.

the_code.mw

I solved an ODE numerically. Code: question_(2).mw

delta*(M(t)*diff~(X, t$2)+C(t)*diff~(X, t))+K(t)*X=f(t)

How to calculate X(t) for different values t and to create a table like following. 

restart;
with(LinearAlgebra):

C:=proc(k,M) local N,P,m,L,T,j,Ld,n,i; 
N:=2^(k-1)*M:
P[0]:= t -> 1:
P[1]:= t -> t:
for m from 2 to M-1 do
   P[m]:= unapply(expand((2*m-1)*t*P[m-1](t)/m - (m-1)*P[m-2](t)/m), t)
end do: 
L:=proc(n,m) local a,b; 
  a := (2*n-2)/2^k; 
  b := 2*n/2^k; 
  return piecewise(a <= t and t <= b, P[m](2^k*t-2*n+1)*sqrt(((m+1)/2)*2^k))
end proc:
T := Vector(N):
for j from 1 to N do T[j] := (j-1/2)/N end do; 
Ld := unapply(Vector(N),t):
for n from 1 to 2^(k-1) do 
  for m from 0 to M-1 do
i := n+2^(k-1)*m:
Ld(t)[i] := L(n,m)
     end do
end do:

return Ld(t):  
end proc:
A:=t->C(2,3);
A(0);
A(0.5);
A(0.3);

Ld is piecewise function depends on t. I want to calculate Ld(0)=A(0), Ld(0.5)=A(0.5) etc.  

What is the problem?