cp's OEIS Frontend

A322214 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.

%I A322214 #9 Dec 04 2018 13:44:50
%S A322214 1,6,-12,-6,6,-12,-12,96,24,-134,-192,114,736,282,-792,-1532,-270,
%T A322214 1932,2004,-96,-3654,-6910,-5532,4836,21500,23454,11850,-8216,-43998,
%U A322214 -57744,-34424,16716,73506,105500,87432,-24474,-230028,-331626,-257616,-163250,316434,852450,1130284,1175748,361110,-652820,-1956330,-2964180,-2922288,-1965174,187806,3863602,6585672,6996900,6199180,366768,-7228866,-14682152,-21063366,-19602108,-10562926,6959976,30061386,50110338,66753126,68131632,37666392
%N A322214 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.
%C A322214 Compare: Product_{n>=1} (1-x^n)^3 = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2).
%H A322214 Paul D. Hanna, <a href="/A322214/b322214.txt">Table of n, a(n) for n = 0..500</a>
%e A322214 G.f.: A(x) = 1 + 6*x - 12*x^2 - 6*x^3 + 6*x^4 - 12*x^5 - 12*x^6 + 96*x^7 + 24*x^8 - 134*x^9 - 192*x^10 + 114*x^11 + 736*x^12 + 282*x^13 - 792*x^14 - 1532*x^15 - 270*x^16 + 1932*x^17 + 2004*x^18 + ...
%e A322214 RELATED SERIES.
%e A322214 The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3 begins
%e A322214 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 A322214 in which this sequence equals the coefficients of x^n*y^n for n >= 0.
%o A322214 (PARI)
%o A322214 {P = prod(n=1, 121, (1 - (x^n + y^n) +O(x^121) +O(y^121))^3 ); }
%o A322214 {a(n) = polcoeff( polcoeff( P, n, x), n, y)}
%o A322214 for(n=0, 120, print1( a(n), ", ") )
%Y A322214 Cf. A322210, A322211, A322213.
%K A322214 sign
%O A322214 0,2
%A A322214 _Paul D. Hanna_, Dec 04 2018