A277292 G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2.
1, 1, 4, 21, 122, 758, 4958, 33509, 233810, 1641150, 12364368, 71807506, 1354944972, -33794258600, 2524565441138, -186642439700891, 16196862324254354, -1602823227559245434, 179707702260054046760, -22656977557634759678794, 3191199098536326709613676, -499206960572108744520132444, 86277300996554233583925645468, -16395890677314419248813441481150
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 +... such that Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2, where A(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 +... A(x) + A(x)^2 = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + 88122*x^16 + 233810*x^17 + 621022*x^18 + 1641150*x^19 + 4411180*x^20 +... Also, A( A(x) + A(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 +... which equals the Catalan series (A000108).
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..201
Programs
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PARI
{a(n) = my(Oxn=x*O(x^(2*n)), A = x +Oxn); for(i=1, 2*n, A = A + (x - subst(A+A^2, x, A-A^2 ))/2); polcoeff(A, 2*n-1)} for(n=1,30,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
(1) A( A(x) + A(x)^2 ) = C(x),
(2) C( A(x) - A(x)^2 ) = A(x),
(3) A( A(x) - A(x)^2 ) = -C(-x),
(4) A( A(x-x^2) + A(x-x^2)^2 ) = x,
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers, A000108.