A382122 G.f. satisfies Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2 and abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

1, 1, 3, 12, 49, 202, 838, 3486, 14575, 60820, 254406, 1061438, 4444802, 18602018, 78066384, 326985608, 1365996909, 5697914836, 23752394338, 99027785702, 413203462516, 1726164299990, 7219911692522, 30228722494504, 126658682953328, 530772842793396, 2224199143900798, 9319843329508200, 39051457052597480

Offset: 0

Author

Paul D. Hanna, Mar 16 2025

Keywords

• nonn

Comments

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

Conjecture: for n > 0, a(n) is odd iff n = 2^k for k >= 0.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 202*x^5 + 838*x^6 + 3486*x^7 + 14575*x^8 + 60820*x^9 + 254406*x^10 + 1061438*x^11 + 4444802*x^12 + ...
Below we illustrate the defining property of this sequence.
The coefficients in 1/A(x)^n begin
 1: [1,  -1, -2,  -7, -24, -84, -298, -1063, ...];
 2: [1,  -2, -3, -10, -30, -92, -283,  -858, ...];
 3: [1,  -3, -3, -10, -24, -57, -119,  -156, ...];
 4: [1,  -4, -2,  -8, -11,  -4,   82,   568, ...];
 5: [1,  -5,  0,  -5,   5,  49,  250,  1060, ...];
 6: [1,  -6,  3,  -2,  21,  90,  348,  1224, ...];
 7: [1,  -7,  7,   0,  35, 112,  364,  1070, ...];
 8: [1,  -8, 12,   0,  46, 112,  304,   672, ...];
 9: [1,  -9, 18,  -3,  54,  90,  186,   135, ...];
10: [1, -10, 25, -10,  60,  48,   35,  -430, ...];
...
The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
 0: [1,  0,  0,  0,  0,   0,   0,    0,    0, ...];
 1: [1,  1,  2,  7, 24,  84, 298, 1063, 3858, ...];
 2: [1,  2,  3, 10, 30,  92, 283,  858, 2646, ...];
 3: [1,  3,  3, 10, 24,  57, 119,  156,  144, ...];
 4: [1,  4,  2,  8, 11,   4,  82,  568, 2578, ...];
 5: [1,  5,  0,  5,  5,  49, 250, 1060, 3800, ...];
 6: [1,  6,  3,  2, 21,  90, 348, 1224, 3654, ...];
 7: [1,  7,  7,  0, 35, 112, 364, 1070, 2394, ...];
 8: [1,  8, 12,  0, 46, 112, 304,  672,  469, ...];
 9: [1,  9, 18,  3, 54,  90, 186,  135, 1629, ...];
10: [1, 10, 25, 10, 60,  48,  35,  430, 3465, ...];
...
the antidiagonals of which add to the Catalan numbers (A000108):
  1 = 1;
  0 + 1 = 1;
  0 + 1 + 1 = 2;
  0 + 2 + 2 + 1 = 5;
  0 + 7 + 3 + 3 + 1 = 14;
  0 + 24 + 10 + 3 + 4 + 1 = 42;
  0 + 84 + 30 + 10 + 2 + 5 + 1 = 132;
  0 + 298 + 92 + 24 + 8 + 0 + 6 + 1 = 429;
  0 + 1063 + 283 + 57 + 11 + 5 + 3 + 7 + 1 = 1430;
  0 + 3858 + 858 + 119 + 4 + 5 + 2 + 7 + 8 + 1 = 4862;
  ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..521

Crossrefs

Cf. A000108, A382123.

Programs

PARI

{a(n) = my(V=[1,1], A, C = (1/x)*serreverse(x - x^2 +x^4*O(x^n)));
for(i=1,n, V = concat(V,'t); A = Ser(V);
V[#V] = 't + polcoef(C - sum(m=1,#V+1, x^m * Ser(abs(Vec( 1/A^m ))) ),#V) );V[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

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

(1) Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2.

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

a(n) ~ c * d^n, where d = 4.1935797816358..., c = 0.142779... - Vaclav Kotesovec, Mar 28 2025