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A229451 G.f.: exp( Sum_{n>=1} (3*n)!/n!^3 * x^n/n ). 3

%I

%S 1,6,63,866,13899,246366,4676768,93322596,1934035965,41286407510,

%T 902562584556,20119266633060,455832458083577,10470568749165246,

%U 243361203186769659,5714294570067499930,135377464019074334826,3232534121305720233264,77726654423445817800164

%N G.f.: exp( Sum_{n>=1} (3*n)!/n!^3 * x^n/n ).

%C The sixth root of the o.g.f. A(x)^(1/6) = 1 + x + 8*x^2 + 101*x^3 + 1569*x^4 + 27445*x^5 + ... appears to have integer coefficients. See A229452. More generally, if A(m,x) := exp( Sum_{n >= 1} (m*n)!/n!^m * x^n/n ), m = 1,2,3,..., then it can be shown that the expansion of A(m,x) has integer coefficients. We conjecture that the expansion of A(m,x)^(1/m!) also has integer coefficients. - _Peter Bala_, Feb 16 2020

%F a(n) ~ c * 3^(3*n)/n^2, where c = 0.406436... - _Vaclav Kotesovec_, Dec 25 2013

%e G.f.: A(x) = 1 + 6*x + 63*x^2 + 866*x^3 + 13899*x^4 + 246366*x^5 +...

%e where

%e log(A(x)) = 6*x + 90*x^2/2 + 1680*x^3/3 + 34650*x^4/4 + 756756*x^5/5 +...+ A006480(n)*x^n/n +...

%t CoefficientList[Series[Exp[6*x*HypergeometricPFQ[{1,1,4/3,5/3},{2,2,2},27*x]],{x,0,20}],x] (* _Vaclav Kotesovec_, Dec 25 2013 *)

%o (PARI) {a(n)=polcoeff(exp(sum(k=1,n,(3*k)!/k!^3*x^k/k) +x*O(x^n)),n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A229452, A006480 (De Bruijn's S(3,n)), A333042, A333043.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 23 2013

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Last modified July 30 19:15 EDT 2021. Contains 346359 sequences. (Running on oeis4.)