A380060 G.f. A(x) satisfies A(x) = Sum_{n=-oo..+oo} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.

1, 2, 4, 16, 66, 284, 1256, 5752, 26944, 128538, 622560, 3053224, 15132720, 75679956, 381430176, 1935464936, 9879523418, 50695803612, 261364074484, 1353156913528, 7032341655200, 36673080496896, 191847911851336, 1006497470833352, 5294344082015344, 27916789231188726, 147534225848777456

Offset

0,2

Comments

The term x^(n^2) * (A - x^n)^n/(1 - x^n*A)^n is invariant under sign reversal of n.

Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.

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 satisfies the following formulas.

(1) A(x) = Sum_{n=-oo..+oo} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.

(2) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.

(3) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * ((1 + A(x))*(1 - x^n)/(1 - x^n*A(x)) - 1)^n.

(4) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * Sum_{k=0..n} binomial(n,k) * (-1)^(n-k) * (1 + A(x))^k*(1 - x^n)^k/(1 - x^n*A(x))^k.

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 66*x^4 + 284*x^5 + 1256*x^6 + 5752*x^7 + 26944*x^8 + 128538*x^9 + 622560*x^10 + ...
where A = A(x) satisfies
A(x) = 1 + 2*x*(A-x)/(1-x*A) + 2*x^4*(A-x^2)^2/(1-x^2*A)^2 + 2*x^9*(A-x^3)^3/(1-x^3*A)^3 + 2*x^16*(A-x^4)^4/(1-x^4*A)^4 + 2*x^25*(A-x^5)^5/(1-x^5*A)^5 + 2*x^36*(A-x^6)^6/(1-x^2*A)^6 + ...
The initial expansions of (A(x) - x^n)^n/(1 - x^n*A(x))^n are
(A - x) / (1 - x*A)   = 1 + 2*x + 8*x^2 + 32*x^3 + 138*x^4 + 616*x^5 + ...
(A-x^2)^2/(1-x^2*A)^2 = 1 + 4*x + 12*x^2 + 56*x^3 + 252*x^4 + 1144*x^5 + ...
(A-x^3)^3/(1-x^3*A)^3 = 1 + 6*x + 24*x^2 + 104*x^3 + 498*x^4 + 2400*x^5 + ...
(A-x^4)^4/(1-x^4*A)^4 = 1 + 8*x + 40*x^2 + 192*x^3 + 952*x^4 + 4784*x^5 + ...
(A-x^5)^5/(1-x^5*A)^5 = 1 + 10*x + 60*x^2 + 320*x^3 + 1690*x^4 + 8892*x^5 + ...
(A-x^6)^6/(1-x^6*A)^6 = 1 + 12*x + 84*x^2 + 496*x^3 + 2796*x^4 + 15456*x^5 + ...
...
SPECIFIC VALUES.
A(t) = 7/4 at t = 0.16664051020540120785482338461092356702146247765117...
A(t) = 5/3 at t = 0.16101585995004397407668302558764589436793162902710...
A(t) = 3/2 at t = 0.14396899091232959619086779782319517939596742329449...
  where 3/2 = 1 + 2*Sum_{n>=1} t^(n^2) * (3/2 - t^n)^n/(1 - t^n*3/2)^n.
A(t) = 4/3 at t = 0.11555123784207763201014608243247194097494223993363...
A(t) = 5/4 at t = 0.09526066252041792904222931736898572772661987990289...
A(t) = 6/5 at t = 0.08064236316571461478644821167332428087306093614749...
A(1/6) = 1.7504486162603029245459813707041863543000995325887...
  where A(1/6) = 1 + 2*Sum_{n>=1} (1/6)^(n^2) * (A(1/6) - (1/6)^n)^n/(1 - (1/6)^n*A(1/6))^n.
A(1/7) = 1.4916405450384080239116001535570642603157648497587...
A(1/8) = 1.3801620298884410186393818492115995979055302865975...
A(1/9) = 1.3133432771808654958611145886546023695475191235704...
A(1/10) = 1.267792455788178105557987788568973110331297169376...

Prog

(PARI) {a(n) = my(A=1); for(k=1, n, A=truncate(A) +x*O(x^k);
A = 1 + 2*sum(m=1, k+1, x^(m^2) * (A - x^m)^m/(1 - x^m*A)^m )); polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380680.

Sequence in context: A175065 A278913 A306979 * A358032 A162119 A213327

Adjacent sequences: A380057 A380058 A380059 * A380061 A380062 A380063

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2025

Status

approved