A001663 -id:A001663 - cp's OEIS frontend

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

N. J. A. Sloane

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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).

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]

Cf. A001663, A001662 (absolute coefficient)

# 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

A380170 Coefficient [k^1] of Wynn's converging polynomial p_n(k) of Weber functions.

Original entry on oeis.org

1, -3, 7, -5, -83, 637, -1425, -22341, 318149, -1415491, -18988393, 444896699, -3268880739, -35114352579, 1317630731647, -14445395761157, -112227733823435, 7047241310852605, -108366459009937881, -487554173851570053, 61301180146129065101, -1271086841777475748099, -1158631507880606959729

Offset: 1

R. J. Mathar, Jan 14 2025

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Corrects a typo (-93) in the version published in 1963, A001663.

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, Table VII in the 2nd part.

Cf. A380169, A001663.

#re-using code of A001664
seq( p(r,1),r=0..12) ;

a(n) = A380169(n,1).

cp's OEIS Frontend

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

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