A266330 Triangle, read by rows, of the coefficients in the g.f.: Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n.

-1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 0, 0, 0, 1, -1, 2, 0, -2, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 3, -1, 1, -3, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 4, 0, 0, 0, -4, 0, 0, 0, 1, -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1, -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1, -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Paul D. Hanna, Dec 27 2015

Keywords

• sign
• tabf

Comments

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Note that the g.f.:

A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n

may be written

A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1)

such that row polynomials R(n,y) consist of square powers of y:

R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2).

Examples

			This triangle of coefficients T(n,k) begins:
  -1, 1;
  -1, 0, 1;
  -1, 1, -1, 1;
  -1, 0, 0, 0, 1;
  -1, 2, 0, -2, 0, 1;
  -1, 0, 0, 0, 0, 0, 1;
  -1, 3, -1, 1, -3, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 4, 0, 0, 0, -4, 0, 0, 0, 1;
  -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1;
  -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1; ...
in which the g.f. of column k > 0 is given by:
z^(k-1)*(1 - z^(k-1))^(k-1) + z^(k*(k+1))/(z^(k+1) - 1)^(k+1).
...
G.f.: A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n may be written as
A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1), where row polynomials R(n,y) consist of square powers of y:
R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2);
this triangle lists the coefficients of y^(k^2) in R(n,y), which begin:
  R(0,y) = y - 1;
  R(1,y) = y^4 - 1;
  R(2,y) = y^9 - y^4 + y - 1;
  R(3,y) = y^16 - 1;
  R(4,y) = y^25 - 2*y^9 + 2*y - 1;
  R(5,y) = y^36 - 1;
  R(6,y) = y^49 - 3*y^16 + y^9 - y^4 + 3*y - 1;
  R(7,y) = y^64 - 1;
  R(8,y) = y^81 - 4*y^25 + 4*y - 1;
  R(9,y) = y^100 + 3*y^16 - 3*y^4 - 1;
  R(10,y) = y^121 - 5*y^36 + 5*y - 1;
  R(11,y) = y^144 - 1;
  R(12,y) = y^169 - 6*y^49 + 6*y^25 - y^16 + y^9 - 6*y^4 + 6*y - 1;
  R(13,y) = y^196 - 1;
  R(14,y) = y^225 - 7*y^64 + 7*y - 1;
  R(15,y) = y^256 + 10*y^36 - 10*y^4 - 1;
  R(16,y) = y^289 - 8*y^81 - 4*y^25 + 4*y^9 + 8*y - 1;
  R(17,y) = y^324 - 1;
  R(18,y) = y^361 - 9*y^100 + 15*y^49 - 15*y^4 + 9*y - 1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..5251, for rows 0..100 in flattened form of this triangle.

Crossrefs

Cf. A217668, A260147 (y=-1).

Programs

PARI

/* Prints rows 0..50 of this triangle: */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
T(n,k)=polcoeff(V[n+1]*y^(n+1) + y*O(y^((n+1)^2)), k^2)}
for(n=0,50,for(k=0,n+1,print1( T(n,k), ", "));print(""))

PARI

/* Quick print of row polynomials (informal): */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
for(n=1,50,print("R("n-1",y) = "V[n]*y^n";")) }

PARI

/* Compare these sums (informal sanity check): */
Axy = sum(n=-16,16, x^n*y^n*(y^n-x^n +O(x^16))^n )
Axy = -(1/y)/(1-x/y) + sum(n=1,15, y^(n^2-1) * ( (x/y)^(n-1)*(1 - (x/y)^(n-1))^(n-1) + (x/y)^(n*(n+1)) / ((x/y)^(n+1) - 1)^(n+1) ) +O(x^16) )

Formula

G.f.: -(1/y)/(1-z) + (1/y) * Sum_{n>=1} y^(n^2) * ( z^(n-1)*(1 - z^(n-1))^(n-1) + z^(n*(n+1)) / (z^(n+1) - 1)^(n+1) ) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n, where z = x/y.

Row sums are all zeros.

Row sums of absolute values of terms yield 2 * A217668(n) for row n>=0.

Sum_{k=0..2*n+1} (-1)^k * T(2*n,k) = (-2) * A260147(n) for n>=0.

Sum_{k=0..2*n+2} (-1)^k * T(2*n+1,k) = 0 for n>=0.

Sum_{k=0..2*n+1} I^(k^2) * T(2*n,k) = (I-1) * A260147(n) for n>=0, where I^2 = -1.

Sum_{k=0..2*n+2} I^(k^2) * T(2*n+1,k) = 0 for n>=0, where I^2 = -1.