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%I #9 Mar 30 2012 18:37:23
%S 1,1,2,5,374,8404542,48017057808567,221378851935038776738734,
%T 1547039418349259734802210245268846703500,
%U 2011481595263302193944358727762275537382584489747186387092683
%N G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k-1,k)^(k^2+k)*x^k]*x^n/n ).
%C Conjecture: this sequence consists entirely of integers.
%C Note that the following g.f. does NOT yield an integer series:
%C . exp( Sum_{n>=1} [Sum_{k>=0} C(n+k-1,k)^(k^2+k-1) * x^k] * x^n/n ).
%F a(n) = (1/n)*Sum_{k=1..n} A183130(k)*a(n-k) for n>0 with a(0) = 1, where A183130(n) = Sum_{k=0..n-1} n*C(n-1,k)^(k^2+k)/(n-k).
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 374*x^4 + 8404542*x^5 +...
%e The logarithm of the g.f. begins:
%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 1475*x^4/4 + 42020826*x^5/5 + 288102296421912*x^6/6 + 1549651963209151973674266*x^7/7 +...
%e and equals the sum of the series:
%e log(A(x)) = (1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 +...)*x
%e + (1 + 2^2*x + 3^6*x^2 + 4^12*x^3 + 5^20*x^4 + 6^30*x^5 +...)*x^2/2
%e + (1 + 3^2*x + 6^6*x^2 + 10^12*x^3 + 15^20*x^4 + 21^30*x^5 +...)*x^3/3
%e + (1 + 4^2*x + 10^6*x^2 + 20^12*x^3 + 35^20*x^4 + 56^30*x^5 +...)*x^4/4
%e + (1 + 5^2*x + 15^6*x^2 + 35^12*x^3 + 70^20*x^4 + 126^30*x^5 +...)*x^5/5
%e + (1 + 6^2*x + 21^6*x^2 + 56^12*x^3 + 126^20*x^4 + 252^30*x^5 +...)*x^6/6 +...
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(k^2+k)*x^k)*x^m/m)+x*O(x^n)),n)}
%o (PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, a(n-k)*sum(j=0, k-1, k*binomial(k-1, j)^(j^2+j)/(k-j))))}
%Y Cf. A183130, A183128.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 25 2010
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