A179277 A(x) = C(x) * C(x^2) * C(x^4) * C(x^8) *...; C = Catalan, A000108.

1, 1, 3, 6, 19, 50, 158, 492, 1635, 5466, 18794, 65332, 230414, 820052, 2945436, 10654808, 38795523, 142045610, 522694866, 1931912036, 7169014298, 26698782108, 99756713732, 373839656616, 1404795235438, 5292114330180, 19982497509316, 75613566762440, 286689890422780

Offset: 0

Author

Gary W. Adamson, Jul 07 2010

Keywords

• nonn

Comments

Let the sequence = A(x) = (1 + x + 3x^2 + 6x^3 + 19x^4 + ...).

Then A(x)/A(x^2) = C(x) = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...).

Examples

			The generating triangle = M:
      1;
      1;
      2,    1;
      5,    1;
     14,    2,   1;
     42,    5,   1;
    132,   14,   2,  1;
    429,   42,   5,  1;
   1430,  132,  14,  2, 1;
   4862,  429,  42,  5, 1;
  16796, 1430, 132, 14, 2, 1;
...
Then take powers of this matrix, obtaining a left-shifted vector considered as a sequence = A179277.
		

Crossrefs

Cf. A000108, A088218.

Programs

Maple

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A179277 := proc(n) if n <= 1 then 1; else add( procname(l)*A000108(n-2*l),l=0..n/2) ; end if; end proc:
seq(A179277(n),n=0..80) ; # R. J. Mathar, Jul 09 2010

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[a[k]*CatalanNumber[n - 2*k], {k, 0, n/2}]; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Nov 27 2024 *)

Formula

Let M = an infinite lower triangular matrix with A000108 in each column but shifted down twice from the previous column, for k>0. Lim_{n->inf.} M^n = A179277, the left shifted vector considered as a sequence: (1 + x + 3x^2 + ...)

a(n) = Sum_{l=0..n/2} a(l)*A000108(n-2*l). - R. J. Mathar, Jul 09 2010

a(n) ~ c * 4^n / n^(3/2), where c = 0.60708656891919662230305917688276343401320432830016456... - Vaclav Kotesovec, Nov 27 2024

Extensions

More terms from R. J. Mathar, Jul 09 2010