A393770 - OEIS

%I #13 Mar 16 2026 04:49:53

%S 1,1,2,5,15,49,168,598,2190,8196,31208,120523,470952,1858620,7397590,

%T 29660563,119689341,485721836,1981073350,8116387242,33387024530,

%U 137840606500,570973544293,2372291510583,9883781844032,41284245752344,172848651689951,725257087734821,3049259724163608,12844360099623312,54199019990087808,229077223340016720

%N G.f.: A(x) = INV(x - x*INV(x - x^3*INV(x - x^5*INV(x - x^7*INV(x - ... - x^(2*n-1)*INV(x - ...)))))), where INV(F(x)) = series reversion of F(x).

%H Paul D. Hanna, <a href="/A393770/b393770.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 4.4379395570069006103370017839662020486541... and c = 0.0779400283864079018838324912195942307... - _Vaclav Kotesovec_, Mar 16 2026

%e G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 + 168*x^7 + 598*x^8 + 2190*x^9 + 8196*x^10 + ...

%e Series_Reversion(A(x)) = x - x*B(x) where

%e B(x) = x + x^4 + 4*x^7 + x^9 + 22*x^10 + 13*x^12 + 140*x^13 + 6*x^14 + 136*x^15 + 970*x^16 + ...

%e Series_Reversion(B(x)) = x - x^3*C(x) where

%e C(x) = x + x^6 + 6*x^11 + x^13 + 51*x^16 + 19*x^18 + 8*x^20 + 506*x^21 + x^22 + 300*x^23 + ...

%e Series_Reversion(C(x)) = x - x^5*D(x) where

%e D(x) = x + x^8 + 8*x^15 + x^17 + 92*x^22 + 25*x^24 + 10*x^26 + x^28 + 1240*x^29 + ...

%e Series_Reversion(D(x)) = x - x^7*E(x) where

%e E(x) = x + x^10 + 10*x^19 + x^21 + 145*x^28 + 31*x^30 + 12*x^32 + x^34 + 2470*x^37 + ...

%e Series_Reversion(E(x)) = x - x^9*F(x) where

%e F(x) = x + x^12 + 12*x^23 + x^25 + 210*x^34 + 37*x^36 + 14*x^38 + x^40 + 4324*x^45 + ...

%e Series_Reversion(F(x)) = x - x^11*G(x) where

%e G(x) = x + x^14 + 14*x^27 + x^29 + 287*x^40 + 43*x^42 + 16*x^44 + x^46 + 6930*x^53 + ...

%o (PARI) {a(n) = my(A=x, m=sqrtint(n+1)); for(k=0, m, A = serreverse(x - x^(2*(m-k)+1)*A + x*O(x^n))); polcoef(GF=A, n)}

%o {upto(n) = a(n); Vec(GF)}

%o upto(30)

%Y Cf. A194956.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 11 2026