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A298692
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G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^( (n+1)^2 ).
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4
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1, 1, 2, 15, 213, 4485, 123566, 4171778, 166069875, 7602292250, 393220294258, 22679300697606, 1443478702575162, 100529312696403699, 7606562231567559478, 621526322941129712986, 54553240678513466719077, 5120001583257750960650134, 511729676123794537164792892, 54270040973557127212080028474, 6087267497390906756985330494931
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OFFSET
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0,3
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COMMENTS
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Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.
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LINKS
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FORMULA
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a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.16989702... - Vaclav Kotesovec, Oct 17 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 213*x^4 + 4485*x^5 + 123566*x^6 + 4171778*x^7 + 166069875*x^8 + 7602292250*x^9 + 393220294258*x^10 + 22679300697606*x^11 + 1443478702575162*x^12 + 100529312696403699*x^13 + 7606562231567559478*x^14 + 621526322941129712986*x^15 + ...
such that
1 = 1/A(x) + C(2,1)/2*x/A(x)^4 + C(6,2)/3*x^2/A(x)^9 + C(12,3)/4*x^3/A(x)^16 + C(20,4)/5*x^4/A(x)^25 + C(30,5)/6*x^5/A(x)^36 + C(42,6)/7*x^6/A(x)^49 + C(56,7)/8*x^7/A(x)^64 + ...
more explicitly,
1 = 1/A(x) + x/A(x)^4 + 5*x^2/A(x)^9 + 55*x^3/A(x)^16 + 969*x^4/A(x)^25 + 23751*x^5/A(x)^36 + 749398*x^6/A(x)^49 + 28989675*x^7/A(x)^64 + ... + A135861(n)*x^n/A(x)^((n+1)^2) + ...
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MATHEMATICA
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terms = 21; A[_] = 1; Do[A[x_] = A[x] - 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/A[x]^((n + 1)^2) + O[x]^(terms), {n, 0, k}], {k, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 06 2018 *)
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = Vec(sum(m=0, #A, binomial(m*(m+1), m)/(m+1) * x^m/Ser(A)^((m+1)^2-1) ))); A[n+1]}
for(n=0, 20, 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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