A384265 G.f. A(x) satisfies A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.

1, 1, 3, 16, 119, 1087, 11408, 132468, 1663047, 22234598, 313303201, 4618133168, 70815362628, 1124901511837, 18450127411436, 311636597558992, 5409374008212747, 96326565666389514, 1757212245656330130, 32798907320986196010, 625759879697614204041, 12192353855092076824051, 242419156191210763668352

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145347 and C(x) = g.f. of A182954 satisfy the following formulas.

(1.a) A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.

(1.b) A(x) = A(x)^2 - x*A( sqrt(x - x/A(x)) )^4.

(2.a) x = Series_Reversion( x/A(x) ) - Series_Reversion( x/A(x)^2 )^2.

(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.

(2.c) B(x) = 1 + x*C(x)^4 where B(x) = A( x*B(x) ) and C(x) = A( x*C(x)^2 ).

(3.a) A(x) = B( x/A(x) ) where B(x/B(x)) = 1 + x*B(x)^3.

(3.b) A(x) = C( x/A(x)^2 ) where C(x) = 1 + x*C(x) * C( x*C(x) )^4.

(4.a) A(x) = A(x)^2 - x*C( x/A(x) )^4.

(4.b) C(x/A(x)) = A( sqrt(x - x/A(x)) ) = A( x/A(x) * C(x/A(x))^2 ).

(4.c) B(x/A(x)^2) = 1 + x*A(x)^2 = A( x/A(x)^2 * B(x/A(x)^2) ).

(5.a) A145347(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.

(5.b) A182954(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

(5.c) A145347(n) = [x^(n-1)] C(x)^4 = [x^(n-1)] A(x)^(2*n+2)*2/(n+1) for n >= 1.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1087*x^5 + 11408*x^6 + 132468*x^7 + 1663047*x^8 + 22234598*x^9 + 313303201*x^10 + ...
 where A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
RELATED SERIES.
The g.f. of A145347 begins
B(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 + 306104*x^7 + 4047988*x^8 + ... + A145347(n)*x^n + ...
 where B(x/B(x)) = 1 + x*B(x)^3
 also, B(x) = A( x*B(x) ).
The g.f. of A182954 begins
C(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 + 710810*x^7 + 10085621*x^8 + ... + A182954(n)*x^n + ...
 where C(x) = 1 + x*C(x) * C( x*C(x) )^4
 also, C(x) = A( x*C(x)^2 ).
C(x)^4 = 1 + 4*x + 26*x^2 + 220*x^3 + 2203*x^4 + 24836*x^5 + 306104*x^6 + ...
 where B(x) = 1 + x*C(x)^4.
RELATED TABLE.
The table of coefficients of x^k in A(x)^n begins
  n = 1: [ 1,  1,   3,   16,   119,   1087, ...];
  n = 2: [ 1, (2),  7,   38,   279,   2508, ...];
  n = 3: [ 1,  3, (12),  67,   489,   4338, ...];
  n = 4: [(1), 4,  18, (104),  759,   6664, ...];
  n = 5: [ 1,  5,  25,  150, (1100),  9586, ...];
  n = 6: [ 1, (6), 33,  206,  1524, (13218),...];
  n = 7: [ 1,  7,  42,  273,  2044,  17689, ...];
  n = 8: [ 1,  8, (52), 352,  2674,  23144, ...];
  n = 9: [ 1,  9,  63,  444,  3429,  29745, ...];
  n =10: [ 1, 10,  75, (550), 4325,  37672, ...];
  n =11: [ 1, 11,  88,  671,  5379,  47124, ...];
  n =12: [ 1, 12, 102,  808, (6609), 58320, ...];
  ...
in which the terms in parenthesis illustrate the property
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
Further, other terms along diagonals obey
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1), and
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + serreverse(x/Ser(A)^2)^2, #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A383563, A145347 (B(x)), A182954 (C(x)).

Sequence in context: A220352 A333682 A125807 * A125222 A199670 A192783

Adjacent sequences: A384262 A384263 A384264 * A384266 A384267 A384268

Keyword

nonn

Author

Paul D. Hanna, May 30 2025

Status

approved