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A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2. 6

%I

%S 1,1,2,6,28,204,2344,43160,1291952,63647664,5218320672,719221578080,

%T 168115994031040,67159892835119296,46166133463916209792,

%U 54941957091151982047616,113826217192695041078973184

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

%F a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) for n>0, with a(0)=1.

%e The recurrence a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) is illustrated by:

%e a(2) = 1*(1) + 3*(1) + 1*(2) = 6;

%e a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;

%e a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.

%e Triangle A022166 begins:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 7, 7, 1;

%e 1, 15, 35, 15, 1;

%e 1, 31, 155, 155, 31, 1;

%e 1, 63, 651, 1395, 651, 63, 1; ...

%t 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-Fran├žois Alcover_, Apr 09 2016 *)

%o (PARI) /* 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)))} /* 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)))} /* Eigensequence at q=2: */ {a(n)=subst(B_q(n),q,2)}

%Y Cf. A022166, A125810, A125811, A125813, A125814, A125815.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 10 2006

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Last modified September 27 03:21 EDT 2020. Contains 337380 sequences. (Running on oeis4.)