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A209442
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G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^k).
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6
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1, 1, 5, 45, 470, 5365, 64766, 813012, 10505163, 138800397, 1866712401, 25470265992, 351717013269, 4906153922941, 69030042202001, 978531875343171, 13961726654230994, 200351151383453293, 2889692388200640136, 41867983817065377259, 609091785100828769195
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OFFSET
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0,3
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COMMENTS
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Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^n * Product_{k=1..n} (1 - 1/G(x)^k)
which holds for all power series G(x) such that G(0)=1.
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LINKS
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FORMULA
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G.f. satisfies: 1+x = A(y) where y = x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+13)/2) * Product_{k=1..n} (A(x)^k - 1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 470*x^4 + 5365*x^5 + 64766*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^2-1)/A(x)^15 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^34 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^45 +...
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MATHEMATICA
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nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^m)/AGF^6, {m, 1, k}], {k, 1, j}], {x, 0, j}]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Dec 01 2014 *)
CoefficientList[1+InverseSeries[Series[x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 01 2014 *)
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PROG
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(PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21 +x^2*O(x^n)), n))}
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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