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A181068 G.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n ). 2

%I

%S 1,1,2,11,80,714,7095,76206,864590,10227727,125001862,1568419058,

%T 20108619244,262510020319,3479914302802,46742907726147,

%U 635092339459857,8716058291255777,120686879727465365,1684357785848110976

%N G.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *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 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...

%e The logarithm begins:

%e log(A(x)) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 + 37632*x^6/6 +...+ A181069(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 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 +...)*x^2/2

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

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

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

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

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

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^4*x^k)*x^m/m)+x*O(x^n)), n)}

%Y Cf. A181069 (log), A181066 (variant).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 08 2010

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Last modified July 23 00:43 EDT 2019. Contains 325228 sequences. (Running on oeis4.)