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A181066 G.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ). 2

%I

%S 1,1,2,7,31,157,865,5051,30774,193669,1250319,8240232,55239187,

%T 375624781,2585449450,17982937876,126222946496,893073250063,

%U 6363674671524,45631735776036,329065051395940,2385126419825231

%N G.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ).

%C Compare g.f. to a g.f. of the Catalan numbers (A000108):

%C . exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^2 *x^k ] *x^n/n ).

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 +...

%e The logarithm begins:

%e log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + 4032*x^6/6 +...+ A181067(n)*x^n/n +...

%e which equals the series:

%e log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x

%e + (1 + 2^3*x + 3^3*x^2 + 4^3*x^3 + 5^3*x^4 + 6^3*x^5 +...)*x^2/2

%e + (1 + 3^3*x + 6^3*x^2 + 10^3*x^3 + 15^3*x^4 + 21^3*x^5 +...)*x^3/3

%e + (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 + 35^3*x^4 + 56^3*x^5 +...)*x^4/4

%e + (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 + 70^3*x^4 + 126^3*x^5 +...)*x^5/5

%e + (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 +...)*x^6/6

%e + (1 + 7^3*x + 28^3*x^2 + 84^3*x^3 + 210^3*x^4 + 462^3*x^5 +...)*x^7/7 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^3*x^k)*x^m/m)+x*O(x^n)), n)}

%Y Cf. A181067 (log), A181068 (variant).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 03 2010

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Last modified August 26 05:50 EDT 2019. Contains 326330 sequences. (Running on oeis4.)