A383377 G.f. satisfies A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

1, 1, 2, 4, 6, 6, 20, 46, 92, 138, 276, 676, 1476, 3332, 5670, 11574, 27262, 61952, 135354, 222848, 549226, 1319282, 3068894, 6449978, 10987080, 27779594, 67311236, 157054012, 313271538, 579149708, 1452091208, 3548249288, 7866783754, 16098393372, 32442930610, 78084645030, 180671169756

Offset

0,3

Comments

Compare to C(x) = Sum_{n>=0} x^n * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Conjecture: a(n) is even for n > 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..515

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = Sum_{n>=0} x^n * abs( 1/A(x)^n ).

(2) a(n) = Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) for n >= 0.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 20*x^6 + 46*x^7 + 92*x^8 + 138*x^9 + 276*x^10 + 676*x^11 + 1476*x^12 + ...
The coefficients in 1/A(x)^n begin
  n = 1: [1, -1, -1, -1,  1,   5,  -11,  -17, ...];
  n = 2: [1, -2, -1,  0,  5,  10,  -33,  -24, ...];
  n = 3: [1, -3,  0,  2,  9,   9,  -70,   -6, ...];
  n = 4: [1, -4,  2,  4, 11,   0, -116,   64, ...];
  n = 5: [1, -5,  5,  5, 10, -16, -160,  210, ...];
  n = 6: [1, -6,  9,  4,  6, -36, -190,  444, ...];
  n = 7: [1, -7, 14,  0,  0, -56, -196,  762, ...];
  n = 8: [1, -8, 20, -8, -6, -72, -172, 1144, ...];
  ...
The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
  n = 0: [1,  0,  0,  0,  0,  0,   0,    0,    0,    0, ...];
  n = 1: [1,  1,  1,  1,  1,  5,  11,   17,    7,   69, ...];
  n = 2: [1,  2,  1,  0,  5, 10,  33,   24,   33,  218, ...];
  n = 3: [1,  3,  0,  2,  9,  9,  70,    6,  123,  377, ...];
  n = 4: [1,  4,  2,  4, 11,  0, 116,   64,  253,  452, ...];
  n = 5: [1,  5,  5,  5, 10, 16, 160,  210,  375,  325, ...];
  n = 6: [1,  6,  9,  4,  6, 36, 190,  444,  399,  102, ...];
  n = 7: [1,  7, 14,  0,  0, 56, 196,  762,  203,  847, ...];
  n = 8: [1,  8, 20,  8,  6, 72, 172, 1144,  349, 1792, ...];
  n = 9: [1,  9, 27, 21,  9, 81, 117, 1557, 1386, 2644, ...];
  n =10: [1, 10, 35, 40,  5, 82,  35, 1960, 3010, 2920, ...];
  ...
in which the antidiagonal sums equal this sequence
  a(0) = 1 = 1;
  a(1) = 0 + 1 = 1;
  a(2) = 0 + 1 + 1 = 2;
  a(3) = 0 + 1 + 2 + 1 = 4;
  a(4) = 0 + 1 + 1 + 3 + 1 = 6;
  a(5) = 0 + 1 + 0 + 0 + 4 + 1 = 6;
  a(6) = 0 + 5 + 5 + 2 + 2 + 5 + 1 = 20;
  a(7) = 0 + 11 + 10 + 9 + 4 + 5 + 6 + 1 = 46;
  a(8) = 0 + 17 + 33 + 9 + 11 + 5 + 9 + 7 + 1 = 92;
  a(9) = 0 + 7 + 24 + 70 + 0 + 10 + 4 + 14 + 8 + 1 = 138;
  a(10) = 0 + 69 + 33 + 6 + 116 + 16 + 6 + 0 + 20 + 9 + 1 = 276;
  ...
illustrating a(n) = Sum_{k=0..n} abs( [x^(n-k)] 1/A(x)^k ) for n >= 0.

Prog

(PARI) {a(n) = my(V=[1], A);
for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = -polcoef(truncate(A) - 1 - sum(m=1, #V+1, x^m * Ser(abs(Vec( 1/A^m ))) ), #V-1) ); V[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A382122.

Sequence in context: A104968 A286894 A346911 * A225187 A281485 A142473

Adjacent sequences: A383374 A383375 A383376 * A383378 A383379 A383380

Keyword

nonn

Author

Paul D. Hanna, May 15 2025

Status

approved