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 A294638 E.g.f. satisfies: A'(x) = A(x) * A(x^2). 2
 1, 1, 1, 3, 9, 33, 153, 963, 6129, 47457, 393489, 3689379, 36673209, 410924097, 4810169961, 64694478627, 878318278497, 13230037503297, 203967546446241, 3494178651687363, 60117798742663401, 1137159539308348641, 21683284489630748601, 452680959717183978243, 9454328250188008785489, 214087305044257976127393, 4862802200825123466537393, 119970186740330465448543843, 2944202974922987534742898329 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..520 FORMULA E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! satisfies: (1) A(x) = exp( Integral A(x^2) dx ). (2) A(x) = 1/A(-x). (3) A(x) = exp( Sum_{n>=0} a(n) * x^(2*n+1) / ((2*n+1)*n!) ) . (4) A(x) = exp( Sum_{n>=0} (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! ). EXAMPLE E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 33*x^5/5! + 153*x^6/6! + 963*x^7/7! + 6129*x^8/8! + 47457*x^9/9! + 393489*x^10/10! + 3689379*x^11/11! + 36673209*x^12/12! + 410924097*x^13/13! + 4810169961*x^14/14! + 64694478627*x^15/15! + 878318278497*x^16/16! + 13230037503297*x^17/17! + 203967546446241*x^18/18! + 3494178651687363*x^19/19! + ... such that A'(x) = A(x) * A(x^2). Also, A(x) = exp( Integral A(x^2) dx ). RELATED SERIES. The logarithm of the e.g.f. is an odd function that begins: log(A(x)) = x + x^3/3 + x^5/(5*2!) + 3*x^7/(7*3!) + 9*x^9/(9*4!) + 33*x^11/(11*5!) + 153*x^13/(13*6!) + 963*x^15/(15*7!) + 6129*x^17/(17*8!) + 47457*x^19/(19*9!) + 393489*x^21/(21*10!) +...+ a(n) * x^(2*n+1)/((2*n+1)*n!) +... which equals Integral A(x^2) dx. Explicitly, log(A(x)) = x + 2*x^3/3! + 12*x^5/5! + 360*x^7/7! + 15120*x^9/9! + 997920*x^11/11! + 101787840*x^13/13! + 16657280640*x^15/15! + 3180450873600*x^17/17! + 837294557299200*x^19/19! +...+ (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! +... PROG (PARI) {a(n) = my(A=1); for(i=1, #binary(n+1), A = exp( intformal( subst(A, x, x^2) +x*O(x^n)) ) ); n!*polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A138292. Sequence in context: A009220 A294035 A007489 * A201968 A264237 A097677 Adjacent sequences:  A294635 A294636 A294637 * A294639 A294640 A294641 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 05 2017 STATUS approved

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Last modified July 18 21:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)