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A292078 G.f. A(x) satisfies: A( x*A(x) - x*A(x)^3 ) = x^2. 2
1, 1, 2, 9, 41, 195, 973, 5063, 27023, 147147, 814736, 4573323, 25964578, 148836155, 860244693, 5007774327, 29335134676, 172795403961, 1022846515313, 6081326577289, 36299846585819, 217453376517864, 1306896005620248, 7877827287152760, 47616211198632181, 288530341065667223, 1752397627719108436, 10666056677548533447, 65048712917677483325, 397445444941987972488, 2432577591939242911470 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Self-convolution forms A272821.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..500

EXAMPLE

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 +..

such that A( x*A(x) - x*A(x)^3 ) = x^2.

RELATED SERIES.

The coefficients of A(x)^2 forms A272821:

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) +...

The cube of the g.f. yields

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 +...

where

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 +...

Define Ai(x) by Ai(A(x)) = x, then:

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 +...

so that A(x) - A(x)^3 = Ai(x^2)/x.

Also, Ai(x) = Ai( Ai(x)^2 ) / (x - x^3).

PROG

(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]}

for(n=1, 40, print1(a(n), ", "))

CROSSREFS

Cf. A272821, A265940.

Sequence in context: A020698 A128752 A074611 * A270766 A181375 A152052

Adjacent sequences:  A292075 A292076 A292077 * A292079 A292080 A292081

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 12 2017

STATUS

approved

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Last modified April 23 21:25 EDT 2019. Contains 322388 sequences. (Running on oeis4.)