login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A322212 G.f.: P(x,y) = Product_{n>=1} (1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as  a square table of coefficients T(n,k) read by antidiagonals. 1
1, -1, -1, -1, 0, -1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 0, -1, -1, -2, -1, -1, -2, -1, -1, 0, 0, -1, 0, 0, 1, -2, 1, 0, 0, -1, 0, 0, -1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1, 0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 0, 0, 0, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 1, 0, 0, -1, 0, -1, 1, 0, -2, 0, 0, 0, 0, -2, 0, 1, -1, 0, -1, 0, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 1, 2, 2, 2, -1, 1, 1, 4, 1, 1, -1, 2, 2, 2, 1, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, -2, -2, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, -1, -2, -1, -2, -2, -2, -2, -2, -1, -2, -1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, -3, -1, 0, 0, 0, 0, -1, -3, 1, 1, 0, 0, 0, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,25

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..7380

EXAMPLE

The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n)) begins

P(x,y) = 1 +  (-1*x - 1*y) + (-1*x^2 + 0*x*y - 1*y^2) + (0*x^3 + 1*x^2*y + 1*x*y^2 + 0*y^3) + (0*x^4 + 1*x^3*y + 0*x^2*y^2 + 1*x*y^3 + 0*y^4) + (1*x^5 + 1*x^4*y + 1*x^3*y^2 + 1*x^2*y^3 + 1*x*y^4 + 1*y^5) + (0*x^6 + 0*x^5*y + 0*x^4*y^2 - 2*x^3*y^3 + 0*x^2*y^4 + 0*x*y^5 + 0*y^6) + (1*x^7 + 0*x^6*y + 0*x^5*y^2 + 0*x^4*y^3 + 0*x^3*y^4 + 0*x^2*y^5 + 0*x*y^6 + 1*y^7) + (0*x^8 - 1*x^7*y + 0*x^6*y^2 - 1*x^5*y^3 - 2*x^4*y^4 - 1*x^3*y^5 + 0*x^2*y^6 - 1*x*y^7 + 0*y^8) + (0*x^9 - 1*x^8*y - 1*x^7*y^2 - 2*x^6*y^3 - 1*x^5*y^4 - 1*x^4*y^5 - 2*x^3*y^6 - 1*x^2*y^7 - 1*x*y^8 + 0*y^9) + (0*x^10 - 1*x^9*y + 0*x^8*y^2 + 0*x^7*y^3 + 1*x^6*y^4 - 2*x^5*y^5 + 1*x^4*y^6 + 0*x^3*y^7 + 0*x^2*y^8 - 1*x*y^9 + 0*y^10) + (0*x^11 - 1*x^10*y - 1*x^9*y^2 + 0*x^8*y^3 - 1*x^7*y^4 + 0*x^6*y^5 + 0*x^5*y^6 - 1*x^4*y^7 + 0*x^3*y^8 - 1*x^2*y^9 - 1*x*y^10 + 0*y^11) + (-1*x^12 - 1*x^11*y + 0*x^10*y^2 + 0*x^9*y^3 - 1*x^8*y^4 + 0*x^7*y^5 + 0*x^6*y^6 + 0*x^5*y^7 - 1*x^4*y^8 + 0*x^3*y^9 + 0*x^2*y^10 - 1*x*y^11 - 1*y^12) + (0*x^13 + 0*x^12*y - 1*x^11*y^2 + 1*x^10*y^3 + 1*x^9*y^4 + 1*x^8*y^5 + 1*x^7*y^6 + 1*x^6*y^7 + 1*x^5*y^8 + 1*x^4*y^9 + 1*x^3*y^10 - 1*x^2*y^11 + 0*x*y^12 + 0*y^13) + (0*x^14 + 0*x^13*y + 1*x^12*y^2 + 2*x^11*y^3 + 2*x^10*y^4 + 2*x^9*y^5 + 3*x^8*y^6 + 2*x^7*y^7 + 3*x^6*y^8 + 2*x^5*y^9 + 2*x^4*y^10 + 2*x^3*y^11 + 1*x^2*y^12 + 0*x*y^13 + 0*y^14) + (-1*x^15 + 0*x^14*y - 1*x^13*y^2 + 1*x^12*y^3 + 0*x^11*y^4 - 2*x^10*y^5 + 0*x^9*y^6 + 0*x^8*y^7 + 0*x^7*y^8 + 0*x^6*y^9 - 2*x^5*y^10 + 0*x^4*y^11 + 1*x^3*y^12 - 1*x^2*y^13 + 0*x*y^14 - 1*y^15) + ...

This square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, ...;

-1, 0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 1, ...;

-1, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 1, -1, 1, 0, ...;

0, 1, 1, -2, 0, -1, -2, 0, 0, 0, 1, 2, 1, 2, 1, 2, ...;

0, 1, 0, 0, -2, -1, 1, -1, -1, 1, 2, 0, 1, 1, 2, 0, ...;

1, 0, 0, -1, -1, -2, 0, 0, 1, 2, -2, 3, 2, 2, 0, -1, ...;

0, 0, 0, -2, 1, 0, 0, 1, 3, 0, 2, 0, -1, -1, -2, 1, ...;

1, -1, -1, 0, -1, 0, 1, 2, 0, 2, 1, 1, 0, -1, -3, 0, ...;

0, -1, 0, 0, -1, 1, 3, 0, 2, 1, 1, -1, -2, -1, -1, -3, ...;

0, -1, -1, 0, 1, 2, 0, 2, 1, 4, -2, -2, 0, -3, -3, -2, ...;

0, -1, 0, 1, 2, -2, 2, 1, 1, -2, -2, 0, -2, -2, -3, -4, ...;

0, -1, -1, 2, 0, 3, 0, 1, -1, -2, 0, 2, -5, -4, -2, -1, ...;

-1, 0, 1, 1, 1, 2, -1, 0, -2, 0, -2, -5, 0, -3, -2, 4, ...;

0, 0, -1, 2, 1, 2, -1, -1, -1, -3, -2, -4, -3, 4, 1, -5, ...;

0, 0, 1, 1, 2, 0, -2, -3, -1, -3, -3, -2, -2, 1, -2, 4, ...;

-1, 1, 0, 2, 0, -1, 1, 0, -3, -2, -4, -1, 4, -5, 4, 4, ...; ...

Alternatively, this sequence can be written as a triangle, starting as

1;

-1, -1;

-1, 0, -1;

0, 1, 1, 0;

0, 1, 0, 1, 0;

1, 1, 1, 1, 1, 1;

0, 0, 0, -2, 0, 0, 0;

1, 0, 0, 0, 0, 0, 0, 1;

0, -1, 0, -1, -2, -1, 0, -1, 0;

0, -1, -1, -2, -1, -1, -2, -1, -1, 0;

0, -1, 0, 0, 1, -2, 1, 0, 0, -1, 0;

0, -1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0;

-1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1;

0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 0;

0, 0, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 1, 0, 0;

-1, 0, -1, 1, 0, -2, 0, 0, 0, 0, -2, 0, 1, -1, 0, -1;

0, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 0;

0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0;

0, 1, 1, 2, 2, 2, -1, 1, 1, 4, 1, 1, -1, 2, 2, 2, 1, 1, 0;

0, 1, 0, 0, 0, 0, -1, 0, -1, -2, -2, -1, 0, -1, 0, 0, 0, 0, 1, 0;

0, 1, 1, 0, 0, -1, -2, -1, -2, -2, -2, -2, -2, -1, -2, -1, 0, 0, 1, 1, 0;

0, 1, 0, 0, 0, 1, 1, -3, -1, 0, 0, 0, 0, -1, -3, 1, 1, 0, 0, 0, 1, 0;

1, 1, 1, -1, 0, 1, -3, 0, -1, -3, -2, 2, -2, -3, -1, 0, -3, 1, 0, -1, 1, 1, 1;

0, 0, 0, -2, -1, -2, -3, -2, -3, -3, -2, -5, -5, -2, -3, -3, -2, -3, -2, -1, -2, 0, 0, 0;

0, 0, 0, -1, -1, -1, -4, -2, -5, -2, -3, -4, 0, -4, -3, -2, -5, -2, -4, -1, -1, -1, 0, 0, 0;

0, 0, 0, -3, -2, -3, -1, -2, -2, -3, -4, -2, -3, -3, -2, -4, -3, -2, -2, -1, -3, -2, -3, 0, 0, 0;

1, 0, 0, -2, -1, -1, -1, -1, -1, -3, 1, -1, -2, 4, -2, -1, 1, -3, -1, -1, -1, -1, -1, -2, 0, 0, 1; ...

PROG

(PARI)

{P = prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) ); }

{T(n, k) = polcoeff( polcoeff( P, n, x), k, y)}

for(n=0, 15, for(k=0, 15, print1( T(n, k), ", ") ); print(""))

CROSSREFS

Cf. A322213.

Sequence in context: A335446 A335460 A086071 * A089813 A343023 A337760

Adjacent sequences:  A322209 A322210 A322211 * A322213 A322214 A322215

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Dec 04 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 13 12:29 EDT 2021. Contains 342936 sequences. (Running on oeis4.)