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Similar to Bell numbers (A000110).
8

%I #36 Sep 18 2024 10:23:46

%S 1,2,6,22,92,426,2150,11708,68282,423948,2788230,19341952,141003552,

%T 1076787624,8589843716,71404154928,617151121998,5535236798058,

%U 51426766394244,494145546973656,4903432458931118,50181840470551778,529009041574922566

%N Similar to Bell numbers (A000110).

%C Equals row sums of triangle A163946. - _Gary W. Adamson_, Aug 06 2009

%H Vincenzo Librandi, <a href="/A091768/b091768.txt">Table of n, a(n) for n = 0..200</a>

%H Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/2003.11533">Wilf equivalences between vincular patterns in inversion sequences</a>, arXiv:2003.11533 [math.CO], 2020.

%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.

%H Zhicong Lin, Sherry H. F. Yan, <a href="https://doi.org/10.1016/j.amc.2019.124672">Vincular patterns in inversion sequences</a>, Applied Mathematics and Computation (2020), Vol. 364, 124672.

%F From _Paul D. Hanna_, Aug 13 2008: (Start)

%F G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x).

%F G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108).

%F 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. (End)

%e The Bell numbers can be generated by;

%e 1

%e 1 2

%e 2 3 5

%e 5 7 10 15

%e 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.

%e This version adds ALL of the entries in the previous column to the new entry.

%e 1

%e 1 2

%e 2 4 6

%e 6 10 16 22

%e where 10=6+2+1+1, 16=10+2+4, 22=16+6

%t 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 *)

%o (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]))

%o (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

%o (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

%Y Close to A074664

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

%K nonn

%O 0,2

%A _Jon Perry_, Mar 06 2004

%E More terms from _Vincenzo Librandi_, Mar 15 2014