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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.

%I A322212 #4 Dec 04 2018 18:40:12
%S A322212 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,
%T A322212 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,
%U A322212 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
%N 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.
%H A322212 Paul D. Hanna, <a href="/A322212/b322212.txt">Table of n, a(n) for n = 0..7380</a>
%e A322212 The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n)) begins
%e A322212 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) + ...
%e A322212 This square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
%e A322212 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, ...;
%e A322212 -1, 0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 1, ...;
%e A322212 -1, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 1, -1, 1, 0, ...;
%e A322212 0, 1, 1, -2, 0, -1, -2, 0, 0, 0, 1, 2, 1, 2, 1, 2, ...;
%e A322212 0, 1, 0, 0, -2, -1, 1, -1, -1, 1, 2, 0, 1, 1, 2, 0, ...;
%e A322212 1, 0, 0, -1, -1, -2, 0, 0, 1, 2, -2, 3, 2, 2, 0, -1, ...;
%e A322212 0, 0, 0, -2, 1, 0, 0, 1, 3, 0, 2, 0, -1, -1, -2, 1, ...;
%e A322212 1, -1, -1, 0, -1, 0, 1, 2, 0, 2, 1, 1, 0, -1, -3, 0, ...;
%e A322212 0, -1, 0, 0, -1, 1, 3, 0, 2, 1, 1, -1, -2, -1, -1, -3, ...;
%e A322212 0, -1, -1, 0, 1, 2, 0, 2, 1, 4, -2, -2, 0, -3, -3, -2, ...;
%e A322212 0, -1, 0, 1, 2, -2, 2, 1, 1, -2, -2, 0, -2, -2, -3, -4, ...;
%e A322212 0, -1, -1, 2, 0, 3, 0, 1, -1, -2, 0, 2, -5, -4, -2, -1, ...;
%e A322212 -1, 0, 1, 1, 1, 2, -1, 0, -2, 0, -2, -5, 0, -3, -2, 4, ...;
%e A322212 0, 0, -1, 2, 1, 2, -1, -1, -1, -3, -2, -4, -3, 4, 1, -5, ...;
%e A322212 0, 0, 1, 1, 2, 0, -2, -3, -1, -3, -3, -2, -2, 1, -2, 4, ...;
%e A322212 -1, 1, 0, 2, 0, -1, 1, 0, -3, -2, -4, -1, 4, -5, 4, 4, ...; ...
%e A322212 Alternatively, this sequence can be written as a triangle, starting as
%e A322212 1;
%e A322212 -1, -1;
%e A322212 -1, 0, -1;
%e A322212 0, 1, 1, 0;
%e A322212 0, 1, 0, 1, 0;
%e A322212 1, 1, 1, 1, 1, 1;
%e A322212 0, 0, 0, -2, 0, 0, 0;
%e A322212 1, 0, 0, 0, 0, 0, 0, 1;
%e A322212 0, -1, 0, -1, -2, -1, 0, -1, 0;
%e A322212 0, -1, -1, -2, -1, -1, -2, -1, -1, 0;
%e A322212 0, -1, 0, 0, 1, -2, 1, 0, 0, -1, 0;
%e A322212 0, -1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0;
%e A322212 -1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1;
%e A322212 0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 0;
%e A322212 0, 0, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 1, 0, 0;
%e A322212 -1, 0, -1, 1, 0, -2, 0, 0, 0, 0, -2, 0, 1, -1, 0, -1;
%e A322212 0, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 0;
%e A322212 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0;
%e A322212 0, 1, 1, 2, 2, 2, -1, 1, 1, 4, 1, 1, -1, 2, 2, 2, 1, 1, 0;
%e A322212 0, 1, 0, 0, 0, 0, -1, 0, -1, -2, -2, -1, 0, -1, 0, 0, 0, 0, 1, 0;
%e A322212 0, 1, 1, 0, 0, -1, -2, -1, -2, -2, -2, -2, -2, -1, -2, -1, 0, 0, 1, 1, 0;
%e A322212 0, 1, 0, 0, 0, 1, 1, -3, -1, 0, 0, 0, 0, -1, -3, 1, 1, 0, 0, 0, 1, 0;
%e A322212 1, 1, 1, -1, 0, 1, -3, 0, -1, -3, -2, 2, -2, -3, -1, 0, -3, 1, 0, -1, 1, 1, 1;
%e A322212 0, 0, 0, -2, -1, -2, -3, -2, -3, -3, -2, -5, -5, -2, -3, -3, -2, -3, -2, -1, -2, 0, 0, 0;
%e A322212 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;
%e A322212 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;
%e A322212 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; ...
%o A322212 (PARI)
%o A322212 {P = prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) ); }
%o A322212 {T(n, k) = polcoeff( polcoeff( P, n, x), k, y)}
%o A322212 for(n=0, 15, for(k=0, 15, print1( T(n, k), ", ") ); print(""))
%Y A322212 Cf. A322213.
%K A322212 sign,tabl
%O A322212 0,25
%A A322212 _Paul D. Hanna_, Dec 04 2018