A385688
E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^3 ).
Original entry on oeis.org
1, 1, 1, 10, 37, 736, 4861, 145552, 1392553, 55772416, 700205401, 35139710464, 546584937229, 32977620613120, 612127803448981, 43150087404292096, 930914421449463505, 75083676142358560768, 1846230024226716759601, 167681514857730519728128, 4629062510444281987051381
Offset: 0
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terms = 21; A[] = 1; Do[A[x] = Exp[x*((A[x] + A[-x])/2)^3] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 07 2025 *)
A385725
E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(i*x) + A(-x) + A(-i*x))/4 ), where i is the imaginary unit.
Original entry on oeis.org
1, 1, 1, 1, 1, 6, 31, 106, 281, 3160, 29701, 176056, 768241, 12702704, 173361371, 1466276176, 8937060081, 195180709248, 3494232292681, 38426220716416, 301057954180801, 8174141246647552, 181144607099402871, 2452803139819922176, 23494461553739152201, 762800754226165963776
Offset: 0
A143598
E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.
Original entry on oeis.org
1, 2, 28, 1176, 103440, 15726880, 3684098496, 1232799974784, 558670427013376, 329559835063067136, 245462725323910487040, 225319148634038399801344, 249936012383478860884217856, 329609037187846742271984869376
Offset: 0
E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +...
A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106:
F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599:
H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...
A263547
E.g.f. satisfies: A(x) = exp( x * real( A(x)^I ) ), where I^2 = -1.
Original entry on oeis.org
1, 1, 1, -2, -11, 36, 421, -1896, -35223, 201232, 5188201, -35856160, -1188970595, 9633456704, 391498316301, -3636762088064, -175238714193967, 1835360835895552, 102369229796454481, -1193179646751072768, -75645902492063337659, 971018266973866894336, 68985480327663686993141, -966900537026209266460672
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! - 2*x^3/3! - 11*x^4/4! + 36*x^5/5! + 421*x^6/6! - 1896*x^7/7! - 35223*x^8/8! + 201232*x^9/9! + 5188201*x^10/10! +...
where
log(A(x)) = x - 3*x^3/3! + 65*x^5/5! - 3787*x^7/7! + 427905*x^9/9! - 79549811*x^11/11! +...+ A036778(n)*x^(2*n-1)/(2*n-1)! +...
which equals Series_Reversion( x/cos(x) ).
Also,
A(x)^I = 1 + I*x - x^2 - 4*I*x^3 + 13*x^4 + 96*I*x^5 - 541*x^6 - 5888*I*x^7/7! + 47545*x^8/8! +...+ A058014(n)*I^n*x^n/n! +...
Further,
Series_Reversion(A(x)-1) = log(1+x)/cos(log(1+x)) = e.g.f. of A009424.
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{a(n) = my(A=1); for(i=1,n+1, A = exp(x*real(A^I) +x*O(x^n))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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{a(n) = my(A=1); for(i=1,n+1, A = exp( serreverse( x/cos(x +x*O(x^n))))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))