cp's OEIS Frontend

A091768 Similar to Bell numbers (A000110).

%I A091768 #36 Sep 18 2024 10:23:46
%S A091768 1,2,6,22,92,426,2150,11708,68282,423948,2788230,19341952,141003552,
%T A091768 1076787624,8589843716,71404154928,617151121998,5535236798058,
%U A091768 51426766394244,494145546973656,4903432458931118,50181840470551778,529009041574922566
%N A091768 Similar to Bell numbers (A000110).
%C A091768 Equals row sums of triangle A163946. - _Gary W. Adamson_, Aug 06 2009
%H A091768 Vincenzo Librandi, <a href="/A091768/b091768.txt">Table of n, a(n) for n = 0..200</a>
%H A091768 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 A091768 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 A091768 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 A091768 From _Paul D. Hanna_, Aug 13 2008: (Start)
%F A091768 G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x).
%F A091768 G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108).
%F A091768 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 A091768 The Bell numbers can be generated by;
%e A091768 1
%e A091768 1 2
%e A091768 2 3 5
%e A091768 5 7 10 15
%e A091768 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 A091768 This version adds ALL of the entries in the previous column to the new entry.
%e A091768 1
%e A091768 1 2
%e A091768 2 4 6
%e A091768 6 10 16 22
%e A091768 where 10=6+2+1+1, 16=10+2+4, 22=16+6
%t A091768 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 A091768 (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 A091768 (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 A091768 (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 A091768 Close to A074664
%Y A091768 Cf. A000110 (Bell Numbers), A033184, A000108, A163946.
%K A091768 nonn
%O A091768 0,2
%A A091768 _Jon Perry_, Mar 06 2004
%E A091768 More terms from _Vincenzo Librandi_, Mar 15 2014