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A322215 G.f.: P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3, 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 A322215 #20 Jan 12 2025 10:50:14
%S A322215 1,-3,-3,0,6,0,5,6,6,5,0,-9,-12,-9,0,0,-9,-6,-6,-9,0,-7,-9,6,-6,6,-9,
%T A322215 -7,0,12,12,27,27,12,12,0,0,12,24,30,6,30,24,12,0,0,12,-12,-23,-24,
%U A322215 -24,-23,-12,12,0,9,12,0,3,-15,-12,-15,3,0,12,9,0,-15,-36,-54,-60,-60,-60,-60,-54,-36,-15,0,0,-15,-24,-23,-30,-9,-12,-9,-30,-23,-24,-15,0,0,-15,-6,-12,51,57,54,54,57,51,-12,-6,-15,0,0,-15,6,24,66,33,69,96,69,33,66,24,6,-15,0,-11,-15,24,49,87,69,127,93,93,127,69,87,49,24,-15,-11
%N A322215 G.f.: P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3, 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.
%C A322215 Compare: Product_{n>=1} (1-x^n)^3 = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2).
%H A322215 Paul D. Hanna, <a href="/A322215/b322215.txt">Table of n, a(n) for n = 0..7380</a>
%e A322215 The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3 begins
%e A322215 P(x,y) = 1 + (-3*x - 3*y) + (0*x^2 + 6*x*y + 0*y^2) + (5*x^3 + 6*x^2*y + 6*x*y^2 + 5*y^3) + (0*x^4 - 9*x^3*y - 12*x^2*y^2 - 9*x*y^3 + 0*y^4) + (0*x^5 - 9*x^4*y - 6*x^3*y^2 - 6*x^2*y^3 - 9*x*y^4 + 0*y^5) + (-7*x^6 - 9*x^5*y + 6*x^4*y^2 - 6*x^3*y^3 + 6*x^2*y^4 - 9*x*y^5 - 7*y^6) + (0*x^7 + 12*x^6*y + 12*x^5*y^2 + 27*x^4*y^3 + 27*x^3*y^4 + 12*x^2*y^5 + 12*x*y^6 + 0*y^7) + (0*x^8 + 12*x^7*y + 24*x^6*y^2 + 30*x^5*y^3 + 6*x^4*y^4 + 30*x^3*y^5 + 24*x^2*y^6 + 12*x*y^7 + 0*y^8) + (0*x^9 + 12*x^8*y - 12*x^7*y^2 - 23*x^6*y^3 - 24*x^5*y^4 - 24*x^4*y^5 - 23*x^3*y^6 - 12*x^2*y^7 + 12*x*y^8 + 0*y^9) + (9*x^10 + 12*x^9*y + 0*x^8*y^2 + 3*x^7*y^3 - 15*x^6*y^4 - 12*x^5*y^5 - 15*x^4*y^6 + 3*x^3*y^7 + 0*x^2*y^8 + 12*x*y^9 + 9*x^0*y^10) + (0*x^11 - 15*x^10*y - 36*x^9*y^2 - 54*x^8*y^3 - 60*x^7*y^4 - 60*x^6*y^5 - 60*x^5*y^6 - 60*x^4*y^7 - 54*x^3*y^8 - 36*x^2*y^9 - 15*x*y^10 + 0*y^11) + (0*x^12 - 15*x^11*y - 24*x^10*y^2 - 23*x^9*y^3 - 30*x^8*y^4 - 9*x^7*y^5 - 12*x^6*y^6 - 9*x^5*y^7 - 30*x^4*y^8 - 23*x^3*y^9 - 24*x^2*y^10 - 15*x*y^11 + 0*y^12) + (0*x^13 - 15*x^12*y - 6*x^11*y^2 - 12*x^10*y^3 + 51*x^9*y^4 + 57*x^8*y^5 + 54*x^7*y^6 + 54*x^6*y^7 + 57*x^5*y^8 + 51*x^4*y^9 - 12*x^3*y^10 - 6*x^2*y^11 - 15*x*y^12 + 0*y^13) + (0*x^14 - 15*x^13*y + 6*x^12*y^2 + 24*x^11*y^3 + 66*x^10*y^4 + 33*x^9*y^5 + 69*x^8*y^6 + 96*x^7*y^7 + 69*x^6*y^8 + 33*x^5*y^9 + 66*x^4*y^10 + 24*x^3*y^11 + 6*x^2*y^12 - 15*x*y^13 + 0*y^14) + (-11*x^15 - 15*x^14*y + 24*x^13*y^2 + 49*x^12*y^3 + 87*x^11*y^4 + 69*x^10*y^5 + 127*x^9*y^6 + 93*x^8*y^7 + 93*x^7*y^8 + 127*x^6*y^9 + 69*x^5*y^10 + 87*x^4*y^11 + 49*x^3*y^12 + 24*x^2*y^13 - 15*x*y^14 - 11*y^15) + ...
%e A322215 This square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
%e A322215   1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -11, ...;
%e A322215   -3, 6, 6, -9, -9, -9, 12, 12, 12, 12, -15, -15, -15, -15, -15, 18, ...;
%e A322215   0, 6, -12, -6, 6, 12, 24, -12, 0, -36, -24, -6, 6, 24, 36, 54, ...;
%e A322215   5, -9, -6, -6, 27, 30, -23, 3, -54, -23, -12, 24, 49, 36, 66, 12, ...;
%e A322215   0, -9, 6, 27, 6, -24, -15, -60, -30, 51, 66, 87, 24, 72, -51, -135, ...;
%e A322215   0, -9, 12, 30, -24, -12, -60, -9, 57, 33, 69, 36, 51, -99, -120, -171, ...;
%e A322215   -7, 12, 24, -23, -15, -60, -12, 54, 69, 127, 21, -3, -141, -192, -192, 3, ...;
%e A322215   0, 12, -12, 3, -60, -9, 54, 96, 93, -66, 69, -213, -189, -201, 24, 15, ...;
%e A322215   0, 12, 0, -54, -30, 57, 69, 93, 24, -18, -204, -234, -150, 51, 36, 174, ...;
%e A322215   0, 12, -36, -23, 51, 33, 127, -66, -18, -134, -285, -165, 93, 171, 309, 629, ...;
%e A322215   9, -15, -24, -12, 66, 69, 21, 69, -204, -285, -192, 180, 69, 228, 621, 240, ...;
%e A322215   0, -15, -6, 24, 87, 36, -3, -213, -234, -165, 180, 114, 285, 819, 111, 480, ...;
%e A322215   0, -15, 6, 49, 24, 51, -141, -189, -150, 93, 69, 285, 736, 156, 396, -592, ...;
%e A322215   0, -15, 24, 36, 72, -99, -192, -201, 51, 171, 228, 819, 156, 282, -528, -618, ...;
%e A322215   0, -15, 36, 66, -51, -120, -192, 24, 36, 309, 621, 111, 396, -528, -792, -1137, ...;
%e A322215  -11, 18, 54, 12, -135, -171, 3, 15, 174, 629, 240, 480, -592, -618, -1137, -1532, ...; ...
%e A322215 Alternatively, this sequence can be written as a triangle, starting as
%e A322215   1;
%e A322215   -3, -3;
%e A322215   0, 6, 0;
%e A322215   5, 6, 6, 5;
%e A322215   0, -9, -12, -9, 0;
%e A322215   0, -9, -6, -6, -9, 0;
%e A322215   -7, -9, 6, -6, 6, -9, -7;
%e A322215   0, 12, 12, 27, 27, 12, 12, 0;
%e A322215   0, 12, 24, 30, 6, 30, 24, 12, 0;
%e A322215   0, 12, -12, -23, -24, -24, -23, -12, 12, 0;
%e A322215   9, 12, 0, 3, -15, -12, -15, 3, 0, 12, 9;
%e A322215   0, -15, -36, -54, -60, -60, -60, -60, -54, -36, -15, 0;
%e A322215   0, -15, -24, -23, -30, -9, -12, -9, -30, -23, -24, -15, 0;
%e A322215   0, -15, -6, -12, 51, 57, 54, 54, 57, 51, -12, -6, -15, 0;
%e A322215   0, -15, 6, 24, 66, 33, 69, 96, 69, 33, 66, 24, 6, -15, 0;
%e A322215   -11, -15, 24, 49, 87, 69, 127, 93, 93, 127, 69, 87, 49, 24, -15, -11;
%e A322215   0, 18, 36, 36, 24, 36, 21, -66, 24, -66, 21, 36, 24, 36, 36, 18, 0;
%e A322215   0, 18, 54, 66, 72, 51, -3, 69, -18, -18, 69, -3, 51, 72, 66, 54, 18, 0;
%e A322215   ...
%o A322215 (PARI)
%o A322215 {P = prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61))^3 ); }
%o A322215 {T(n, k) = polcoeff( polcoeff( P, n, x), k, y)}
%o A322215 for(n=0, 15, for(k=0, 15, print1( T(n, k), ", ") ); print(""))
%Y A322215 Cf. A322214 (main diagonal), A322216 (antidiagonal sums).
%K A322215 sign,tabl,look
%O A322215 0,2
%A A322215 _Paul D. Hanna_, Dec 04 2018