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A298696
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G.f.: Sum_{n>=0} binomial(n*(n+1), n)/(n+1) * x^n / (1 + x)^(n*(n+1)).
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4
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1, 1, 3, 28, 410, 8386, 220962, 7140736, 273712896, 12146997564, 612813677300, 34647736132384, 2170381958609592, 149223874286440552, 11173356309069883320, 905099760309260722560, 78870011549256151244288, 7356892186010414244194704, 731435433368215011644979504, 77216368897429504869064200256, 8626428901029156775683110378400, 1016792561657783042048699052986016
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OFFSET
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0,3
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COMMENTS
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Compare g.f. 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)))) = 6.176554609483480358231680164050876553672889... and c = 0.226094037474708064867716267720651240574569526310006844420310030408773601638... - Vaclav Kotesovec, Feb 07 2018
c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) * sqrt(2) * (2 + LambertW(-2*exp(-2))) / (4*Pi*sqrt(-LambertW(-2*exp(-2))) * sqrt(1 + LambertW(-2*exp(-2)))). - Vaclav Kotesovec, Mar 18 2022
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 28*x^3 + 410*x^4 + 8386*x^5 + 220962*x^6 + 7140736*x^7 + 273712896*x^8 + 12146997564*x^9 + 612813677300*x^10 + ...
such that
A(x) = 1 + C(2,1)/2*x/(1+x)^2 + C(6,2)/3*x^2/(1+x)^6 + C(12,3)/4*x^3/(1+x)^12 + C(20,4)/5*x^4/(1+x)^20 + C(30,5)/6*x^5/(1+x)^30 + ...
more explicitly,
A(x) = 1 + x/(1+x)^2 + 5*x^2/(1+x)^6 + 55*x^3/(1+x)^12 + 969*x^4/(1+x)^20 + 23751*x^5/(1+x)^30 + ... + A135861(n)*x^n/(1+x)^(n*(n+1)) + ...
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MATHEMATICA
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terms = 22; s = Sum[Binomial[n*(n + 1), n]/(n + 1)*x^n/(1 + x)^(n*(n + 1)), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Feb 05 2018 *)
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n, binomial(m*(m+1), m)/(m+1)*x^m/(1+x +x*O(x^n))^(m*(m+1)) ) ); polcoeff(A, n)}
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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