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%I #32 Sep 12 2017 23:20:54
%S 1,1,2,9,41,195,973,5063,27023,147147,814736,4573323,25964578,
%T 148836155,860244693,5007774327,29335134676,172795403961,
%U 1022846515313,6081326577289,36299846585819,217453376517864,1306896005620248,7877827287152760,47616211198632181,288530341065667223,1752397627719108436,10666056677548533447,65048712917677483325,397445444941987972488,2432577591939242911470
%N G.f. A(x) satisfies: A( x*A(x) - x*A(x)^3 ) = x^2.
%C Self-convolution forms A272821.
%H Paul D. Hanna, <a href="/A292078/b292078.txt">Table of n, a(n) for n = 1..500</a>
%e G.f.: A(x) = x + x^3 + 2*x^5 + 9*x^7 + 41*x^9 + 195*x^11 + 973*x^13 + 5063*x^15 + 27023*x^17 + 147147*x^19 + 814736*x^21 + 4573323*x^23 + 25964578*x^25 + 148836155*x^27 + 860244693*x^29 + 5007774327*x^31 + 29335134676*x^33 + 172795403961*x^35 +..
%e such that A( x*A(x) - x*A(x)^3 ) = x^2.
%e RELATED SERIES.
%e The coefficients of A(x)^2 forms A272821:
%e A(x)^2 = x^2 + 2*x^4 + 5*x^6 + 22*x^8 + 104*x^10 + 508*x^12 + 2581*x^14 + 13590*x^16 + 73255*x^18 + 402096*x^20 + 2240803*x^22 + 12645756*x^24 + 72120577*x^26 + 415017628*x^28 + 2406756231*x^30 +...+ A272821(n)*x^(2*n) +...
%e The cube of the g.f. yields
%e A(x)^3 = x^3 + 3*x^5 + 9*x^7 + 40*x^9 + 195*x^11 + 978*x^13 + 5063*x^15 + 27009*x^17 + 147147*x^19 + 814782*x^21 + 4573323*x^23 + 25964403*x^25 + 148836155*x^27 + 860245368*x^29 + 5007774327*x^31 + 29335131945*x^33 + 172795403961*x^35 +...
%e where
%e A(x) - A(x)^3 = x - x^5 + x^9 - 5*x^13 + 14*x^17 - 46*x^21 + 175*x^25 - 675*x^29 + 2731*x^33 - 11323*x^37 + 47642*x^41 - 203618*x^45 +...
%e Define Ai(x) by Ai(A(x)) = x, then:
%e Ai(x) = x - x^3 + x^5 - 5*x^7 + 14*x^9 - 46*x^11 + 175*x^13 - 675*x^15 + 2731*x^17 - 11323*x^19 + 47642*x^21 - 203618*x^23 + 879655*x^25 - 3834767*x^27 + 16853975*x^29 - 74585691*x^31 + 332094891*x^33 - 1486751803*x^35 + 6688440045*x^37 +...
%e so that A(x) - A(x)^3 = Ai(x^2)/x.
%e Also, Ai(x) = Ai( Ai(x)^2 ) / (x - x^3).
%o (PARI) {a(n) = my(A=[1,0,1]); for(i=1, n, A=concat(A, [0,0]); F=x*Ser(A); A[#A] = -Vec( subst(F, x, x*F - x*F^3 ) )[#A] ); A[2*n-1]}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A272821, A265940.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 12 2017
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