A136247 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial h(n,x) with h(0,x)=1, h(1,x)=1-x and recursively h(n,x) = 1 + n -(1-x)*(1-h(n-1,x)) - n*h(n-2,x).

1, 1, -1, 1, -1, 1, 1, 2, 2, -1, 1, 6, -4, -3, 1, 1, -4, -20, 6, 4, -1, 1, -40, 8, 44, -8, -5, 1, 1, -12, 188, -6, -80, 10, 6, -1, 1, 308, 136, -546, -10, 130, -12, -7, 1, 1, 416, -1864, -628, 1256, 50, -196, 14, 8, -1, 1, -2664, -3640, 6696, 1984, -2506, -126, 280, -16, -9, 1

Offset: 0

Author

Roger L. Bagula, Mar 17 2008

Keywords

• easy
• tabl
• sign

Comments

Row sums are 1, 0, 1, 4, 1, -14, 1, 106, 1, -944, 1, 10396, 1, -135134, 1, 2027026, 1, -34459424, 1, 654729076, 1...

[Row sums s(n) appear to obey s(n) -2*s(n-1) +(n+1)*s(n-2) +2*(1-n)*s(n-3) +(n-2)* s(n-4)=0. - R. J. Mathar, Dec 04 2011]

Examples

			1;
1, -1;
1, -1, 1;
1, 2, 2, -1;
1, 6, -4, -3, 1;
1, -4, -20,6, 4, -1;
1, -40, 8, 44, -8, -5, 1;
1, -12, 188, -6, -80,10, 6, -1;
1, 308, 136, -546, -10, 130, -12, -7, 1;
1, 416, -1864, -628, 1256, 50, -196, 14, 8, -1;
1, -2664, -3640, 6696, 1984, -2506, -126,280, -16, -9, 1;
		

References

• Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.

Crossrefs

Cf. A137286.

Programs

Maple

h := proc(n,x)
    if n = 0 then
        1 ;
    elif n = 1 then
        1-x ;
    else
        1+n-(1-x)*(1-procname(n-1,x)) -n*procname(n-2,x) ;
        expand(%) ;
    end if;
end proc:
A136247 := proc(n,k)
    coeftayl(h(n,x),x=0,k) ;
end proc:
seq(seq(A136247(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 04 2011

Mathematica

Clear[h, a, n, x, y, c, d] (*Solve linear Shabat transform for Hermite type recursion*) Solve[c*x0 + d - x*(c*x1 + d) + n*(c*x2 + d) == 0, x0] c = -1; d = 1; Solve[y = c*x + d == 0, x] h[x, 0] = 1; h[x, 1] = 1 - x; h[x_, n_] := h[x, n] = -(-1 - n + (1 - x) - (1 - x)* h[ x, n - 1] + n *h[x, n - 2]); Table[ExpandAll[h[x, n]], {n, 0, 10}]; a = Table[CoefficientList[h[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[h[x, n], x]], {n, 0, 10}];