A277293 -id:A277293 - cp's OEIS frontend

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

Paul D. Hanna, Oct 12 2016

sign

			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).

Paul D. Hanna, Table of n, a(n) for n = 1..201

Cf. A318008, A277293, A277294, A000108, A179270, A277292.

{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),", "))

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.

A277294 G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4.

Original entry on oeis.org

1, 2, 35, 812, 21359, 623244, 18568947, 638475040, 13249877870, 2024051330358, -355660668390645, 130426094235366208, -54120354853298252400, 27045033537893084984896, -15918675371944450999486319, 10905983125914263654567255488, -8603776324190250513027830925715, 7743542274281960968631431340349870, -7886327135586560316787947739703112447, 9023297352140462809043434127286176617288, -11524615288427474577090651960651636283169590

Offset: 1

Paul D. Hanna, Oct 12 2016

sign

			G.f.: A(x) = x + 2*x^7 + 35*x^13 + 812*x^19 + 21359*x^25 + 623244*x^31 + 18568947*x^37 + 638475040*x^43 + 13249877870*x^49 + 2024051330358*x^55 +...
such that Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4, where
A(x)^4 = x^4 + 8*x^10 + 164*x^16 + 4120*x^22 + 113970*x^28 + 3416128*x^34 + 104776764*x^40 + 3565389600*x^46 + 88390775151*x^52 +...
A(x) + A(x)^4 = x + x^4 + 2*x^7 + 8*x^10 + 35*x^13 + 164*x^16 + 812*x^19 + 4120*x^22 + 21359*x^25 + 113970*x^28 + 623244*x^31 + 3416128*x^34 + 18568947*x^37 + 104776764*x^40 + 638475040*x^43 + 3565389600*x^46 + 13249877870*x^49 + 88390775151*x^52 + 2024051330358*x^55 +...
Also,
A( A(x) + A(x)^4 ) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 + 969*x^16 + 7084*x^19 + 53820*x^22 +...+ binomial(4*n-3,n-1)/(4*n-3)*x^(3*n-2) +...
which is a g.f. of A002293.
Note that the following is an integer series:
sqrt(A(x)/x) = 1 + x^6 + 17*x^12 + 389*x^18 + 10146*x^24 + 294863*x^30 + 8741468*x^36 + 301536587*x^42 + 6008625027*x^48 + 994498807123*x^54 - 179176440388960*x^60 + 65367342797524884*x^66 - 27123073712119583646*x^72 +...

Cf. A277292, A277293, A002293.

{a(n) = my(Oxn=x*O(x^(6*n)), A = x +Oxn); for(i=1, 6*n, A = A + (x - subst(A + A^4, x, A - A^4 ))/2); polcoeff(A, 6*n-5)}
for(n=1,25,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n) * x^(6*n-5) satisfies:
(1) A( A(x) + A(x)^4 ) = G(x),
(2) G( A(x) - A(x)^4 ) = A(x),
(3) A( A(x) - A(x)^4 ) = -G(-x),
(4) A( A(x-x^4) + A(x-x^4)^4 ) = x,
where G(x) = x + G(x)^4 = Sum_{n>=1} binomial(4*n-3,n-1)/(4*n-3) * x^(3*n-2).

cp's OEIS Frontend

A277292 G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2.

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