A193442
E.g.f.: exp( Sum_{n>=1} x^(2*n)/A000108(n) ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where A000108 is the Catalan numbers.
Original entry on oeis.org
1, 2, 24, 624, 27744, 1857600, 173256192, 21357471744, 3350185712640, 649812612225024, 152385461527633920, 42429768718712094720, 13819620038445598408704, 5199913478124022299033600, 2236448840442614178722611200, 1089467246881095674146487009280
Offset: 0
E.g.f.: A(x) = 1 + 2*x^2/2! + 24*x^4/4! + 624*x^6/6! + 27744*x^8/8! + 1857600*x^10/10! + 173256192*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
where
log(A(x)) = x^2 + x^4/2 + x^6/5 + x^8/14 + x^10/42 + x^12/132 + x^14/429 + x^16/1430 +...+ (n+1)*x^(2*n)/C(2*n,n) +...
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{a(n)=(2*n)!*polcoeff(exp(sum(m=1,n,(m+1)*x^(2*m)/binomial(2*m,m))+O(x^(2*n+1))),2*n)}
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/* Using formula for e.g.f. = exp(L(x)): */
{a(n)=local(Ox=O(x^(2*n+1)), L=-1 + 2*(8+x^2)/(4-x^2 +Ox)^2 + 24*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^5)); (2*n)!*polcoeff(exp(L), 2*n)}
A193443
E.g.f.: exp( Sum_{n>=1} x^(2*n)/(2*A000108(n)) ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where A000108 is the Catalan numbers.
Original entry on oeis.org
1, 1, 9, 177, 6081, 320625, 23901993, 2382903873, 305213701185, 48729724204833, 9471295552801545, 2198860046959656465, 600311814859681301889, 190227653770262659729425, 69194247433728324962214825, 28616922449430718198313413665, 13345389352004839017903164910465
Offset: 0
E.g.f.: A(x) = 1 + x^2/2! + 9*x^4/4! + 177*x^6/6! + 6081*x^8/8! + 320625*x^10/10! + 23901993*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
where
log(A(x)) = x^2/2 + x^4/4 + x^6/10 + x^8/28 + x^10/84 + x^12/264 + x^14/858 + x^16/2860 +...+ (n+1)*x^(2*n)/(2*C(2*n,n)) +...
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{a(n)=(2*n)!*polcoeff(exp(sum(m=1,n,(m+1)*x^(2*m)/binomial(2*m,m)/2)+O(x^(2*n+1))),2*n)}
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/* Using formula for e.g.f. = exp(L(x)): */
{a(n)=local(Ox=O(x^(2*n+1)),L=-1/2 + (8+x^2)/(4-x^2 +Ox)^2 + 12*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^5)); (2*n)!*polcoeff(exp(L),2*n)}
A193444
E.g.f.: exp( Sum_{n>=1} n!*x^(2*n)/(2*n)! ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.
Original entry on oeis.org
1, 1, 5, 51, 857, 21045, 702597, 30379839, 1642718865, 108171928521, 8495805003525, 782625366185355, 83400601634195049, 10163125433194019325, 1402348965454767334725, 217258436356989650347095, 37513434482581646048138145, 7172552434209226974773867025
Offset: 0
E.g.f.: A(x) = 1 + x^2/2! + 5*x^4/4! + 51*x^6/6! + 857*x^8/8! + 21045*x^10/10! + 702597*x^12/12! +...+ a(n)*x^n/(2*n)! +...
where
log(A(x)) = x^2/2! + 2!*x^4/4! + 3!*x^6/6! + 4!*x^8/8! + 5!*x^10/10! +...
Showing 1-3 of 3 results.
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