A392210 G.f.: x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + sq(x/(1-4*x) + ...)))) where sq(x) = x^2, an infinitely nested square.

1, 1, 4, 18, 96, 579, 3834, 27404, 209052, 1688389, 14350934, 127790843, 1187887298, 11493457661, 115475260028, 1202307540187, 12950284520902, 144086601275823, 1653734575825448, 19556088359856621, 238010195665296028, 2978302718258295407, 38282183452466374942, 505012097362619472074

Offset

1,3

Comments

This sequence equals the self-convolution (shifted right) of the binomial transform of this sequence.

It appears that a(2*n-1) is even for n > 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = x + A(x/(1-x))^2.

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

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

(4) A(x/(1-x^2)) = x/(1-x^2) + A(x/(1-x-x^2))^2.

(5) A(x/(1-n*x)) = x/(1-n*x) + A( x/(1-(n+1)*x) )^2 for all n.

(6) A(x/(1-n*x^2)) = x/(1-n*x^2) + A( x/(1 - x - n*x^2) )^2 for all n.

(7) A(x/g(x)) = x/g(x) + A( x/(g(x) - x) )^2 for all g(x) such that g(0) = 1.

(8) A(x) = x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + sq(x/(1-4*x) + ...)))) where sq(x) = x^2, an infinitely nested square.

a(n) = Sum_{k=1..n-1} b(n-k) * b(k) for n > 1 with a(1) = 1 where b(n) = Sum_{k=1..n} binomial(n-1,k-1) * a(k).

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 18*x^4 + 96*x^5 + 579*x^6 + 3834*x^7 + 27404*x^8 + 209052*x^9 + 1688389*x^10 + ... where A(x) = x + A(x/(1-x))^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 44*x^5 + 244*x^6 + 1494*x^7 + 9918*x^8 + 70564*x^9 + 533644*x^10 + ... which equals A(x/(1+x)) - x/(1+x).
A(x/(1-x)) = x + 2*x^2 + 7*x^3 + 34*x^4 + 197*x^5 + 1285*x^6 + 9175*x^7 + 70483*x^8 + 575909*x^9 + ... which equals x/(1-x) + A(x/(1-2*x))^2.
A(x/(1-x))^2 = x^2 + 4*x^3 + 18*x^4 + 96*x^5 + 579*x^6 + 3834*x^7 + 27404*x^8 + 209052*x^9 + ... which equals A(x) - x.
RELATED TABLES.
The table of coefficients of x^k, k>=1, in A(x/(1-n*x)) for n >= 0 begins
  n = 0: [1, 1,  4,  18,   96,   579,   3834, ...];
  n = 1: [1, 2,  7,  34,  197,  1285,   9175, ...];
  n = 2: [1, 3, 12,  62,  384,  2691,  20638, ...];
  n = 3: [1, 4, 19, 108,  717,  5367,  43983, ...];
  n = 4: [1, 5, 28, 178, 1280, 10243,  89410, ...];
  n = 5: [1, 6, 39, 278, 2181, 18729, 174079, ...];
  n = 6: [1, 7, 52, 414, 3552, 32835, 325350, ...]; ...
The table of coefficients of x^k, k>=1, in A(x/(1-(n+1)*x))^2 for n >= 0 begins
  n = 0: [0, 1,  4,  18,   96,   579,   3834, ...];
  n = 1: [0, 1,  6,  33,  196,  1284,   9174, ...];
  n = 2: [0, 1,  8,  54,  368,  2659,  20574, ...];
  n = 3: [0, 1, 10,  81,  636,  5124,  43254, ...];
  n = 4: [0, 1, 12, 114, 1024,  9219,  85314, ...];
  n = 5: [0, 1, 14, 153, 1556, 15604, 158454, ...];
  n = 6: [0, 1, 16, 198, 2256, 25059, 278694, ...]; ...
Note that the differences between row n of both tables are the powers of n.

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = x + subst(A, x, x/(1-x + x*O(x^n)))^2 ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A090363, A119685, A000108, A393172.

Sequence in context: A180567 A378733 A379281 * A152392 A001563 A094304

Adjacent sequences: A392207 A392208 A392209 * A392211 A392212 A392213

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2026

Status

approved