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A322214
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a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.
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3
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1, 6, -12, -6, 6, -12, -12, 96, 24, -134, -192, 114, 736, 282, -792, -1532, -270, 1932, 2004, -96, -3654, -6910, -5532, 4836, 21500, 23454, 11850, -8216, -43998, -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
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OFFSET
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0,2
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COMMENTS
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Compare: Product_{n>=1} (1-x^n)^3 = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2).
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LINKS
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EXAMPLE
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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 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3 begins
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) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.
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PROG
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(PARI)
{P = prod(n=1, 121, (1 - (x^n + y^n) +O(x^121) +O(y^121))^3 ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, 120, print1( a(n), ", ") )
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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