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A294474
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G.f. A(x) satisfies: Product_{n=-oo..+oo} 1 + x^n*(1 + A(x)^n)^n = 4.
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1
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1, -1, -7, 5, 87, -189, -1941, 6515, 49795, -229867, -1343239, 8320303, 36790861, -305098413, -989510467, 11255950785, 24982028631, -415990506193, -537407696757, 15351256446099, 6424844980489, -564158722662119, 245084502553583, 20595643407110741, -26734732721361955, -744981949442089559, 1643062426565260297, 26621246008843146659, -85112137855646858055, -936300170489142867171, 4053227035994699790679, 32249453191213868198529
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies: P(x) * Q(x) = 4 where
P(x) = Product_{n>=0} ( 1 + x^n*(1 + A(x)^n)^n ),
Q(x) = Product_{n>=1} ( 1 + A(x)^(n^2)/(x + x*A(x)^n)^n ).
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EXAMPLE
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G.f.: A(x) = x - x^2 - 7*x^3 + 5*x^4 + 87*x^5 - 189*x^6 - 1941*x^7 + 6515*x^8 + 49795*x^9 - 229867*x^10 - 1343239*x^11 + 8320303*x^12 + 36790861*x^13 - 305098413*x^14 - 989510467*x^15 + 11255950785*x^16 + 24982028631*x^17 - 415990506193*x^18 - 537407696757*x^19 + 15351256446099*x^20 + 6424844980489*x^21 - 564158722662119*x^22 + 245084502553583*x^23 + 20595643407110741*x^24 - 26734732721361955*x^25 +...
such that P(x) * Q(x) = 4 where
P(x) = 2 * (1 + x*(1 + A(x))) * (1 + x^2*(1 + A(x)^2)^2) * (1 + x^3*(1 + A(x)^3)^3) * (1 + x^4*(1 + A(x)^4)^4) * (1 + x^5*(1 + A(x)^5)^5) *...
Q(x) = (1 + A(x)/(x + x*A(x))) * (1 + A(x)^4/(x + x*A(x)^2)^2) * (1 + A(x)^9/(x + x*A(x)^3)^3) * (1 + A(x)^16/(x + x*A(x)^4)^4) *...
Explicitly,
P(x) = 2 + 2*x + 4*x^2 + 2*x^3 - 4*x^4 + 12*x^5 + 120*x^6 - 352*x^7 - 2980*x^8 + 11680*x^9 + 78820*x^10 - 402092*x^11 - 2127062*x^12 + 14447232*x^13 + 57933462*x^14 - 527035730*x^15 - 1535125698*x^16 + 19361173486*x^17 + 37513015102*x^18 - 712839675304*x^19 - 740477725406*x^20 + 26211270910440*x^21 + 4739879806196*x^22 - 959773087051380*x^23 + 640308620069676*x^24 + 34904312707737186*x^25 +...
Q(x) = 2 - 2*x - 2*x^2 + 4*x^3 + 6*x^4 - 28*x^5 - 100*x^6 + 642*x^7 + 2322*x^8 - 18850*x^9 - 56244*x^10 + 594794*x^11 + 1327778*x^12 - 19779660*x^13 - 28527454*x^14 + 677346604*x^15 + 440164160*x^16 - 23558817688*x^17 + 3440836740*x^18 + 825188815716*x^19 - 792149118816*x^20 - 28930022822608*x^21 + 52487049980348*x^22 + 1010034689733148*x^23 - 2767421447992010*x^24 - 34946816786828024*x^25 +...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); G = x*Ser(A); A[#A] = -Vec(prod(m=-#A-1, #A+1, 1 + x^m*(1 + G^m)^m ))[#A]/2); A[n]}
for(n=1, 40, 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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