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A153851 Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3. 6
1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448, 810629805, 15938815794, 326653743510, 6949638584208, 153009877730525, 3477623225388063, 81429702521625843, 1961136442605508341, 48513571089988199157 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..18.

FORMULA

G.f.: A(x) = Sum_{n>=0} a(2n+1)*x^(2n+1) = Series_Reversion[x - A(x)^3].

EXAMPLE

G.f.: A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...

A(x - A(x)^3) = x where

A(x)^3 = x^3 + 3*x^5 + 21*x^7 + 208*x^9 + 2517*x^11 + 34851*x^13 +...

SYSTEM OF RELATED FUNCTIONS.

A = A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:

A = 1 + x^2*B^3;

B = A + x^2*C^3;

C = B + x^2*D^3;

D = C + x^2*E^3;

E = D + x^2*F^3; ...

where the functions xB, xC, xD, etc., are successive iterations of A(x):

x*A = A(x),

x*B = A(A(x)) = g.f. of A153852,

x*C = A(A(A(x))) = g.f. of A153853,

x*D = A(A(A(A(x)))) = g.f. of A153854, etc.

The nonzero coefficients of these functions begin:

A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448,...];

B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374,...];

C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479,...];

D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067,...];

E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, 3866962617,...];

F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, 8466942393,...];

G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, 17278647147,...];

H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, 33260962995,..]; ...

The main diagonal in the above table is A153850.

PROG

(PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x-subst(A^3, x, x+x^2*O(x^(2*n))))) ; polcoeff(A, 2*n-1)}

CROSSREFS

Cf. A153852, A153853, A153854, A153850; variant: A139702.

Sequence in context: A207412 A324447 A060435 * A141372 A306030 A152170

Adjacent sequences:  A153848 A153849 A153850 * A153852 A153853 A153854

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 21 2009

STATUS

approved

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)