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 A249934 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(3*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)). 4
 1, 1, 1, 4, 19, 107, 671, 4600, 34218, 276415, 2439426, 23724674, 256361107, 3091554768, 41560590331, 618957882104, 10119509431084, 179887355572358, 3446915545155744, 70686674091569072, 1542478858735415921, 35650141769790146478, 869385516566240903091, 22299067147713040916568 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Compare the g.f. to the identity: G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1)) which holds for all power series G(x) such that G(0)=1. LINKS Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..240 (first 100 terms from Paul D. Hanna) FORMULA G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+3)) * Product_{k=1..n} (A(x)^(2*k-1) - 1). a(n) ~ exp(Pi^2/24) * 12^n * n^(n-1) / (sqrt(6) * exp(n) * Pi^(2*n-1)). - Vaclav Kotesovec, Dec 01 2014 EXAMPLE A(x) = 1 + x + x^2 + 4*x^3 + 19*x^4 + 107*x^5 + 671*x^6 + 4600*x^7 + 34218*x^8 +... The g.f. satisfies: x = (A(x)-1)/A(x)^4 + (A(x)-1)*(A(x)^3-1)/A(x)^10 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^18 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^28 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^40 +... 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^(2m-1))/AGF^3, {m, 1, k}], {k, 1, j}], {x, 0, j}]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* More efficient than PARI program, Vaclav Kotesovec, Nov 30 2014 *) PROG (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)^(3*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A214692. Sequence in context: A088129 A082030 A052751 * A174992 A182541 A241839 Adjacent sequences:  A249931 A249932 A249933 * A249935 A249936 A249937 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 27 2014 STATUS approved

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Last modified February 21 05:23 EST 2020. Contains 332086 sequences. (Running on oeis4.)