A127782 G.f. satisfies A(x) = 1 + x*A(x+x^2).

1, 1, 1, 2, 4, 11, 33, 114, 438, 1845, 8458, 41823, 221539, 1250269, 7481758, 47278652, 314374316, 2192798077, 16000160519, 121831654450, 965946444587, 7958739329386, 68023023892680, 602115897105136, 5511499584735858

Offset: 0

Author

Paul D. Hanna, Jan 28 2007

Keywords

• nonn

Comments

Equals eigensequence of triangle A026729. - Gary W. Adamson, Jan 16 2009

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

Examples

			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
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500
• Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Crossrefs

Cf. A026729, A000110, A360885, A360886, A360894.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*binomial(n-i, i-1), i=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

Mathematica

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 *)
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 *)

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)

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

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 */

Formula

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]

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

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