A091768 Similar to Bell numbers (A000110).

1, 2, 6, 22, 92, 426, 2150, 11708, 68282, 423948, 2788230, 19341952, 141003552, 1076787624, 8589843716, 71404154928, 617151121998, 5535236798058, 51426766394244, 494145546973656, 4903432458931118, 50181840470551778, 529009041574922566

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

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). a(n) = A000108(n) + Sum_{k=0..n-1} a(k)*C(2*n-k-1,n-k-1)*(k+2)/(n+1) for n>=0; eigensequence (shift left) of the Catalan triangle A033184. - Paul D. Hanna, Aug 13 2008

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

Close to A074664

Cf. A000110 (Bell Numbers), A033184, A000108, A163946.

Sequence in context: A124295 A074664 A367442 * A229741 A261518 A185349

Adjacent sequences: A091765 A091766 A091767 * A091769 A091770 A091771

Keyword

nonn

Author

Jon Perry, Mar 06 2004

Extensions

More terms from Vincenzo Librandi, Mar 15 2014

Status

approved