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

%I A179277 #17 Nov 27 2024 10:53:32
%S A179277 1,1,3,6,19,50,158,492,1635,5466,18794,65332,230414,820052,2945436,
%T A179277 10654808,38795523,142045610,522694866,1931912036,7169014298,
%U A179277 26698782108,99756713732,373839656616,1404795235438,5292114330180,19982497509316,75613566762440,286689890422780
%N A179277 A(x) = C(x) * C(x^2) * C(x^4) * C(x^8) *...; C = Catalan, A000108.
%C A179277 Let the sequence = A(x) = (1 + x + 3x^2 + 6x^3 + 19x^4 + ...).
%C A179277 Then A(x)/A(x^2) = C(x) = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...).
%F A179277 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 + ...)
%F A179277 a(n) = Sum_{l=0..n/2} a(l)*A000108(n-2*l). - _R. J. Mathar_, Jul 09 2010
%F A179277 a(n) ~ c * 4^n / n^(3/2), where c = 0.60708656891919662230305917688276343401320432830016456... - _Vaclav Kotesovec_, Nov 27 2024
%e A179277 The generating triangle = M:
%e A179277       1;
%e A179277       1;
%e A179277       2,    1;
%e A179277       5,    1;
%e A179277      14,    2,   1;
%e A179277      42,    5,   1;
%e A179277     132,   14,   2,  1;
%e A179277     429,   42,   5,  1;
%e A179277    1430,  132,  14,  2, 1;
%e A179277    4862,  429,  42,  5, 1;
%e A179277   16796, 1430, 132, 14, 2, 1;
%e A179277 ...
%e A179277 Then take powers of this matrix, obtaining a left-shifted vector considered as a sequence = A179277.
%p A179277 A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A179277 A179277 := proc(n) if n <= 1 then 1; else add( procname(l)*A000108(n-2*l),l=0..n/2) ; end if; end proc:
%p A179277 seq(A179277(n),n=0..80) ; # _R. J. Mathar_, Jul 09 2010
%t A179277 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 *)
%Y A179277 Cf. A000108, A088218.
%K A179277 nonn
%O A179277 0,3
%A A179277 _Gary W. Adamson_, Jul 07 2010
%E A179277 More terms from _R. J. Mathar_, Jul 09 2010