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
Keywords
Examples
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
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.
Programs
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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 *) -
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
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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
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).
Extensions
More terms from Vincenzo Librandi, Mar 15 2014
Comments