A153851 Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3.

1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448, 810629805, 15938815794, 326653743510, 6949638584208, 153009877730525, 3477623225388063, 81429702521625843, 1961136442605508341, 48513571089988199157

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:

(1) A(x) = Series_Reversion( x - A(x)^3 ).

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n) / n!. - Paul D. Hanna, Sep 07 2020

(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x / n! ). - Paul D. Hanna, Sep 07 2020

(4) x = A(A( x-x^3 - A(x)^3 )). - Paul D. Hanna, Sep 07 2020

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; variants: A139702, A213591.

Sequence in context: A362167 A324447 A060435 * A141372 A306030 A152170

Adjacent sequences: A153848 A153849 A153850 * A153852 A153853 A153854

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2009

Status

approved