OFFSET
0,2
COMMENTS
Equals row sums of triangle A163946. - Gary W. Adamson, Aug 06 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
FORMULA
From Paul D. Hanna, Aug 13 2008: (Start)
G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x).
G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108).
EXAMPLE
The Bell numbers can be generated by;
1
1 2
2 3 5
5 7 10 15
where the Bell numbers are the last entry on each line. This last entry is the first entry on the next line and then the last two entries of the previous column are added, e.g. 7=5+2, 10=7+3, 15=10+5.
This version adds ALL of the entries in the previous column to the new entry.
1
1 2
2 4 6
6 10 16 22
where 10=6+2+1+1, 16=10+2+4, 22=16+6
MATHEMATICA
nmax=21; b = ConstantArray[0, nmax]; b[[1]]=1; Do[b[[n+1]] = Binomial[2*n, n]/(n+1) + Sum[b[[k+1]]*Binomial[2*n-k-1, n-k-1]*(k+2)/(n+1), {k, 0, n-1}], {n, 1, nmax-1}]; b (* Vaclav Kotesovec, Mar 13 2014 *)
PROG
(PARI) v=vector(20); for (i=1, 20, v[i]=vector(i)); v[1][1]=1; for (i=2, 20, v[i][1]=v[i-1][i-1]; for (j=2, i, v[i][j]=v[i][j-1]+sum(k=j-1, i-1, v[k][j-1]))); for (i=1, 20, print1(", "v[i][i]))
(PARI) a(n)=binomial(2*n, n)/(n+1)+sum(k=0, n-1, a(k)*binomial(2*n-k-1, n-k-1)*(k+2)/(n+1)) \\ Paul D. Hanna, Aug 13 2008
(PARI) a(n)=local(A=1+x*O(x^n), C=serreverse(x-x^2+x^2*O(x^n))/x); for(i=0, n, A=C+x*C^2*subst(A, x, x*C)); polcoeff(A, n) \\ Paul D. Hanna, Aug 13 2008
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Jon Perry, Mar 06 2004
EXTENSIONS
More terms from Vincenzo Librandi, Mar 15 2014
STATUS
approved