A168407
E.g.f.: Sum_{n>=0} (exp(2^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).
Original entry on oeis.org
1, 2, 20, 712, 91920, 44874784, 85939843136, 660213878210688, 20540390859740217600, 2592165941692975372042752, 1324271564605167892188248409088, 2730585827960928853182474922961668096
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 712*x^3/3! + 91920*x^4/4! +...
A(x) = 1 + (exp(2*x) - 1) + (exp(4*x) - 1)^2/2! + (exp(8*x) - 1)^3/3! +...+ (exp(2^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(2^n) where Bell(x) = exp(exp(x)-1):
Bell(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...+ A000110(n)*x^n/n! +...
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Table[BellB[n, 2^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)
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{a(n)=local(infnty=n^4+10);round(exp(-2^n)*sum(k=0,infnty,(2^k*k)^n/k!))}
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{a(n)=n!*polcoeff(sum(k=0,n,(exp(2^k*x +x*O(x^n))-1)^k/k!),n)}
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{a(n)=n!*polcoeff(exp(2^n*(exp(x +x*O(x^n))-1)),n)}
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{S2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)}
{a(n)=sum(k=0,n,S2(n,k)*2^(n*k))}
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{a(n)=polcoeff(sum(k=0,n,(2^k*x)^k/prod(j=1,k,1-j*2^k*x+x*O(x^n))),n)}
A168404
E.g.f.: Sum_{n>=0} tan(2^n*x)^n/n!.
Original entry on oeis.org
1, 2, 16, 528, 67584, 34210304, 69391122432, 565356426987520, 18478277930015260672, 2419401354886413876592640, 1267940756758206239694099841024, 2658665157828553829995392867121496064
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 528*x^3/3! + 67584*x^4/4! +...
A(x) = 1 + tan(2*x) + tan(4*x)^2/2! + tan(8*x)^3/3! + tan(16*x)^4/4! +...+ tan(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(tan(x)):
G(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 37*x^5/5! + 177*x^6/6! +...+ A006229(n)*x^n/n! +...
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{a(n)=n!*polcoeff(sum(k=0,n,tan(2^k*x +x*O(x^n))^k/k!),n)}
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{a(n)=n!*polcoeff(exp(2^n*tan(x +x*O(x^n))),n)}
A168405
E.g.f.: Sum_{n>=0} arcsin(2^n*x)^n/n!.
Original entry on oeis.org
1, 2, 16, 520, 66560, 33882400, 69055283200, 564153087455360, 18462510039810703360, 2418626471936038215754240, 1267795676362601991645220044800, 2658560574070850656450883768752998400
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! + ...
A(x) = 1 + arcsin(2*x) + arcsin(4*x)^2/2! + arcsin(8*x)^3/3! + arcsin(16*x)^4/4! + ... + arcsin(2^n*x)^n/n! + ...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arcsin(x)):
G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 20*x^5/5! + 85*x^6/6! + ... + A006228(n)*x^n/n! + ...
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{a(n)=n!*polcoeff(sum(k=0,n,asin(2^k*x +x*O(x^n))^k/k!),n)}
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{a(n)=n!*polcoeff(exp(2^n*asin(x +x*O(x^n))),n)}
A168406
E.g.f.: Sum_{n>=0} arctan(2^n*x)^n/n!.
Original entry on oeis.org
1, 2, 16, 496, 63488, 32899840, 68049141760, 560546415810560, 18415229458563727360, 2416302337337071616327680, 1267360474688679165942982246400, 2658246833688954938616062542151680000
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 496*x^3/3! + 63488*x^4/4! + ...
A(x) = 1 + arctan(2*x) + arctan(4*x)^2/2! + arctan(8*x)^3/3! + arctan(16*x)^4/4! + ... + arctan(2^n*x)^n/n! + ...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arctan(x)):
G(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! + 5*x^5/5! + 145*x^6/6! + ... + A002019(n)*x^n/n! + ...
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{a(n)=n!*polcoeff(sum(k=0,n,atan(2^k*x +x*O(x^n))^k/k!),n)}
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{a(n)=n!*polcoeff(exp(2^n*atan(x +x*O(x^n))),n)}
Showing 1-4 of 4 results.