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 A181998 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^k). 6
 1, 1, 3, 18, 124, 935, 7443, 61510, 522467, 4532452, 39985628, 357641094, 3235846003, 29565353095, 272429349163, 2528938553028, 23629834081955, 222080711420655, 2098112946860819, 19915641133236764, 189853287434733709, 1816924035668823659, 17450483777418686431 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS FORMULA G.f. satisfies: 1+x = A(y) where y = x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10. G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+9)/2) * Product_{k=1..n} (A(x)^k - 1). EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 124*x^4 + 935*x^5 + 7443*x^6 +... The g.f. satisfies: x = (A(x)-1)/A(x)^5 + (A(x)-1)*(A(x)^2-1)/A(x)^11 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^18 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^26 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^35 +... MATHEMATICA 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^4, {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 - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 01 2014 *) PROG (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10 +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)^(4*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A001002, A181997, A209441, A209442, A214694 (variant). Sequence in context: A074564 A108241 A199421 * A176277 A289429 A004987 Adjacent sequences:  A181995 A181996 A181997 * A181999 A182000 A182001 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 05 2012 STATUS approved

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Last modified November 23 09:38 EST 2017. Contains 295115 sequences.