A159597
G.f.: A(x) = exp( Sum_{n>=1} [ D^n x/(1-x)^3 ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 7, 37, 245, 2094, 24661, 410376, 9809637, 334520167, 16192227784, 1107914634442, 106788033119369, 14525652771018918, 2780328926392863928, 751651711717655433750, 286240041470280077141769
Offset: 0
G.f.: A(x) = 1 + x + 7*x^2 + 37*x^3 + 245*x^4 + 2094*x^5 +...
log(A(x)) = Sum_{n>=1} [x + 2^n*3*x^2 + 3^n*6*x^3 +...]^n/n.
D^n x/(1-x)^3 = x + 2^n*3*x^2 + 3^n*6*x^3 + 4^n*10*x^4 +...
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^m*k*(k+1)/2*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
A159598
G.f.: A(x) = exp( Sum_{n>=1} [ D^n x(1+x)/(1-x)^3 ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 9, 52, 389, 3741, 49908, 938799, 25477165, 984680146, 54180019253, 4211350678751, 462028240134476, 71561459522839253, 15611478225943599423, 4816139618587302209166, 2092942812095475521879845
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 52*x^3 + 389*x^4 + 3741*x^5 +...
log(A(x)) = Sum_{n>=1} [x + 2^(n+2)*x^2 + 3^(n+2)*x^3 +...]^n/n.
D^n x(1+x)/(1-x)^2 = x + 2^(n+2)*x^2 + 3^(n+2)*x^3 + 4^(n+2)*x^4 +...
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^(m+2)*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
A159599
E.g.f.: A(x) = exp( Sum_{n>=1} [ D^n exp(x) ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 4, 27, 304, 5685, 177486, 9305821, 807656872, 113141689065, 25091265489130, 8644033129800321, 4584172093683770820, 3704744323753306881229, 4538175408875808587259022, 8381136688938251234193247485
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 304*x^4/4! +...
log(A(x)) = x + 3*x^2/2! + 17*x^3/3! + 190*x^4/4! + 3889*x^5/5! +...
log(A(x)) = (D^1 e^x) + (D^2 e^x)^2/2 + (D^3 e^x)^3/3 +...
D^1 exp(x) = (1)*x*exp(x);
D^2 exp(x) = (1 + x)*x*exp(x);
D^3 exp(x) = (1 + 3*x + x^2)*x*exp(x);
D^4 exp(x) = (1 + 7*x + 6*x^2 + x^3)*x*exp(x);
D^5 exp(x) = (1 + 15*x + 25*x^2 + 10*x^3 + x^4)*x*exp(x); ...
D^n exp(x) = n-th iteration of operator D = x*d/dx on exp(x) equals the g.f. of the n-th row of triangle A008277 (S2(n,k)) times x*exp(x), and so is related to the n-th Bell number.
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n, sum(k=1,n,k^m*x^k/k!+x*O(x^n))^m/m))); n!*polcoeff(A,n)}
Showing 1-3 of 3 results.