%I #35 Oct 05 2024 22:13:53
%S 1,1,1,2,4,11,33,114,438,1845,8458,41823,221539,1250269,7481758,
%T 47278652,314374316,2192798077,16000160519,121831654450,965946444587,
%U 7958739329386,68023023892680,602115897105136,5511499584735858
%N G.f. satisfies A(x) = 1 + x*A(x+x^2).
%C Equals eigensequence of triangle A026729. - _Gary W. Adamson_, Jan 16 2009
%C In Barry[2011] on page 9 is Example 12 where the first column of the eigentriangle of the skew binomial matrix is this sequence. - _Michael Somos_, Oct 03 2024
%H Vaclav Kotesovec, <a href="/A127782/b127782.txt">Table of n, a(n) for n = 0..500</a>
%H Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
%F a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k,k) * a(n-1-k) for n>0 with a(0)=1. [corrected by _Seiichi Manyama_, Feb 25 2023]
%F a(n) ~ c * Bell(n) * LambertW(n) / (n*exp(LambertW(n)^2/2)), where c = 1.93210869..., or a(n) ~ c * exp(n/LambertW(n) - LambertW(n)^2/2 - 1 - n) * n^(n-1) / (LambertW(n)^(n-1) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Mar 12 2014
%F a(n) = Sum_{k=0..n-1} binomial(k, n-k-1) * a(k) for n>0 with a(0)=1. (from Barry[2011]) - _Michael Somos_, Oct 03 2024
%e G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 33*x^6 + 114*x^7 + ... - _Michael Somos_, Oct 03 2024
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(a(n-i)*binomial(n-i, i-1), i=1..n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 11 2016
%t nmax = 20; b = ConstantArray[0, nmax]; b[[1]] = 1; Do[b[[n+2]] = Sum[Binomial[n-k, k]*b[[n-k+1]], {k, 0, n}], {n, 0, nmax-2}]; b (* _Vaclav Kotesovec_, Mar 12 2014 *)
%t a[ n_] := If[n<1, Boole[n==0], a[n] = Sum[Binomial[k, n-1-k] * a[k], {k, 0, n-1}]]; (* _Michael Somos_, Oct 03 2024 *)
%o (PARI) a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+x*subst(A,x,x+x^2)); polcoeff(A,n)
%o (PARI) a(n)=if(n==0, 1, sum(k=0, (n-1)\2,binomial(n-1-k,k)*a(n-1-k))); \\ corrected by _Seiichi Manyama_, Feb 25 2023
%o (PARI) {a(n) = my(A = 1 + O(x)); for(k=1, n, A = 1 + x*subst(A, x, x+x^2)); polcoeff(A, n)}; /* _Michael Somos_, Oct 03 2024 */
%Y Cf. A026729, A000110, A360885, A360886, A360894.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 28 2007