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URL: https://oeis.org/A295767

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A295767
G.f. A(x) satisfies: A(x + A(x)*A(-x)) = x - A(x)*A(-x).
1
1, 2, 4, 18, 76, 500, 2888, 23018, 160556, 1449996, 11575640, 114832932, 1019757080, 10926139752, 106088136208, 1215141302498, 12753198909052, 155094128725196, 1745058840478104, 22420718376535948, 268759075046461512, 3634051693946151736, 46176378783947578800, 655022571579520952068, 8785797027703008422264, 130388708648538590304216, 1839515449214236524003120
OFFSET
1,2
LINKS
FORMULA
G.f. A(x) satisfies:
(1) A(x) = x + 2*B( (A(x) + x)/2 ), where B(x) = -A(x)*A(-x).
(2) A(x) = -x + 2*Series_Reversion( x + A(x)*A(-x) ).
(3) x = A( -x + 2*Series_Reversion( x - A(x)*A(-x) ) ).
(4) A(x + A(x)*A(-x)) = x - A(x)*A(-x).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 18*x^4 + 76*x^5 + 500*x^6 + 2888*x^7 + 23018*x^8 + 160556*x^9 + 1449996*x^10 + 11575640*x^11 + 114832932*x^12 + 1019757080*x^13 + 10926139752*x^14 + 106088136208*x^15 + 1215141302498*x^16 +...
such that A(x + A(x)*A(-x)) = x - A(x)*A(-x).
RELATED SERIES.
-A(x)*A(-x) = x^2 + 4*x^4 + 96*x^6 + 4060*x^8 + 239920*x^10 + 17996072*x^12 + 1630314752*x^14 + 173202828908*x^16 + 21167253920784*x^18 + 2935439183937720*x^20 +...
(A(x) + x)/2 = x + x^2 + 2*x^3 + 9*x^4 + 38*x^5 + 250*x^6 + 1444*x^7 + 11509*x^8 + 80278*x^9 + 724998*x^10 + 5787820*x^11 + 57416466*x^12 +...
sqrt( -A(x)*A(-x) ) = x + 2*x^3 + 46*x^5 + 1938*x^7 + 115026*x^9 + 8678836*x^11 + 790630586*x^13 + 84398006438*x^15 + 10355026866054*x^17 + 1440596696075200*x^19 + 224937262867609220*x^21 +...
PROG
(PARI) {a(n) = my(A=[1], G=x); for(i=1, n, A=concat(A, 0); G = x*Ser(A); A[#A] = -Vec(subst(G, x, x + G*subst(G, x, -x)) + G*subst(G, x, -x))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
Cf. A141202.
Sequence in context: A139104 A014448 A277033 * A318230 A075836 A295370
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 04 2017
STATUS
approved