cp's OEIS Frontend

A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

%I A125812 #26 Feb 21 2025 20:36:39
%S A125812 1,1,2,6,28,204,2344,43160,1291952,63647664,5218320672,719221578080,
%T A125812 168115994031040,67159892835119296,46166133463916209792,
%U A125812 54941957091151982047616,113826217192695041078973184,412563248965919999955196308224,2627807814905396804499456018866688
%N A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.
%H A125812 Alois P. Heinz, <a href="/A125812/b125812.txt">Table of n, a(n) for n = 0..90</a>
%F A125812 a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) for n>0, with a(0)=1.
%F A125812 a(n) = Sum_{k>=0} 2^k * A125810(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A125812 The recurrence a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) is illustrated by:
%e A125812 a(2) = 1*(1) + 3*(1) + 1*(2) = 6;
%e A125812 a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;
%e A125812 a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.
%e A125812 Triangle A022166 begins:
%e A125812 1;
%e A125812 1, 1;
%e A125812 1, 3, 1;
%e A125812 1, 7, 7, 1;
%e A125812 1, 15, 35, 15, 1;
%e A125812 1, 31, 155, 155, 31, 1;
%e A125812 1, 63, 651, 1395, 651, 63, 1; ...
%p A125812 b:= proc(o, u, t) option remember;
%p A125812      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(2^(u+j-1)*
%p A125812         b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
%p A125812     end:
%p A125812 a:= n-> b(n, 0$2):
%p A125812 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 21 2025
%t A125812 a[0] = 1; a[n_] := a[n] = Sum[QBinomial[n-1, k, 2] a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 09 2016 *)
%o A125812 (PARI)
%o A125812 /* q-Binomial coefficients: */ {C_q(n,k)=if(n<k || k<0,0,if(n==0 || k==0,1,prod(j=n-k+1,n,1-q^j)/prod(j=1,k,1-q^j)))}
%o A125812 /* q-Bell numbers = eigensequence of q-binomial triangle: */ {B_q(n)=if(n==0,1,sum(k=0,n-1,B_q(k)*C_q(n-1,k)))}
%o A125812 /* Eigensequence at q=2: */ {a(n)=subst(B_q(n),q,2)}
%Y A125812 Cf. A022166, A125810, A125811, A125813, A125814, A125815.
%Y A125812 Column k=2 of A381369.
%K A125812 nonn
%O A125812 0,3
%A A125812 _Paul D. Hanna_, Dec 10 2006