A001664 Quadratic coefficient of the n-th converging polynomial of Weber functions.

1, -6, 25, -60, -203, 3710, -21347, -50400, 2465969, -24201342, -14909791, 4154706556, -61829802067, 107889525510, 13926895008805, -296622934827816, 1387504872714793, 80367331405832714, -2381736125794455767, 19480923855903871284, 721535152036700012069, -29550684521199839783538

Offset: 2

Author

N. J. A. Sloane

Keywords

• sign

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument, part Ia, part Ib, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-754 (two parts). In (45) the factor p_{r-2,2} should read p_{r-2,1}.
• P. Wynn, Converging factors for the Weber parabolic cylinder functions ... [Annotated scan of part 2 only]

Crossrefs

Cf. A001663, A001662 (absolute coefficient)

Programs

Maple

# equation (47)
prs := proc(r,k)
    if r = 0  then
        1 ;
    elif r = 1  then
        -1+k ; # (38)
    elif r =2 then
        1-3*k+k^2 ;
    end if;
end proc:
p := proc(r,s)
    option remember ;
    local k,a,lambda,mu,phi,theta ;
    # theta := 0 ; # valid for Table VII
    phi :=1 : # (30) for theta=0
    a := 1/2 ; # specific in Table VII
    lambda := 2*(a-1) ; # (15)
    mu := (a-1/2)*(a-3/2) ; # (13)
    if r = s then
        return 1; # eq (42)
    elif s > r or s <0 then
        return 0 ;
    elif r <=2 then
        coeff(prs(r,k),k,s) ;
    elif s = 0 then
        # eq (46)
        2*(phi+2)*procname(r,1) -8*procname(r,2)
        +4*(4*r-lambda-2)*procname(r-1,1)
        +2*(lambda*(phi+1)-2*(r-1)*phi-4*r)*procname(r-1,0)
        -4*(mu-2*lambda*(r-1)+4*(r-1)^2)*procname(r-2,0) ;
        return %/(phi+1) ;
    elif s = 1 then
        # eq (45)
        # note that the 2nd index of the last p is wrong in the publication
        4*(phi+2)*procname(r,2) -24*procname(r,3) # unreadable index is 3
        +8*(4*r-lambda-2)*procname(r-1,2)
        -8*procname(r-1,1)+2*(phi+2)*procname(r-1,0)
        +2*(lambda*(phi+1)-2*(r-1)*phi-4*r)*procname(r-1,1)
        -4*(lambda-4*r+4)*procname(r-2,0)
        -4*(mu-2*lambda*(r-1)+4*(r-1)^2)*procname(r-2,1) ;
        return %/(phi+1) ;
    elif s= r-1 then
        # eq (43)
        2*(phi+2)*r*procname(r,r) -8*(r-1)*procname(r-1,r-1)
        +2*(phi+2)*procname(r-1,r-2)+2*(lambda*(phi+1)-2*(r-1)*phi-4*r)*procname(r-1,r-1)
        -4*procname(r-2,r-3)-4*(lambda-4*r+4)*procname(r-2,r-2) ;
        return %/(phi+1) ;
    else
        # eq (44)
        2*(s+1)*(phi+2)*procname(r,s+1) -4*(s+1)*(s+2)*procname(r,s+2)
        +4*(4*r-lambda-2)*(s+1)*procname(r-1,s+1)-8*s*procname(r-1,s)
        +2*(phi+2)*procname(r-1,s-1)+2*(lambda*(phi+1)-2*(r-1)*phi-4*r)*procname(r-1,s)
        -4*procname(r-2,s-2)-4*(lambda-4*r+4)*procname(r-2,s-1)
        -4*(mu-2*lambda*(r-1)+4*(r-1)^2)*procname(r-2,s) ;
        return %/(phi+1) ;
    end if;
end proc:
A001664 := proc(n)
    p(n,2) ;
end proc:
seq(A001664(n),n=2..30) ; # R. J. Mathar, Jan 13 2025