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A125814 q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4. 5

%I

%S 1,1,2,8,72,1552,84416,12107584,4726583424,5150624868864,

%T 16010990175691264,144648776120641766400,3857411545088966609514496,

%U 307705704204270334224705015808,74294186209325019487040708053442560

%N q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4.

%F a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) for n>0, with a(0)=1.

%e The recurrence: a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k)

%e is illustrated by:

%e a(2) = 1*(1) + 5*(1) + 1*(2) = 8;

%e a(3) = 1*(1) + 21*(1) + 21*(2) + 1*(8) = 72;

%e a(4) = 1*(1) + 85*(1) + 357*(2) + 85*(8) + 1*(72) = 1552.

%e Triangle A022168 begins:

%e 1;

%e 1, 1;

%e 1, 5, 1;

%e 1, 21, 21, 1;

%e 1, 85, 357, 85, 1;

%e 1, 341, 5797, 5797, 341, 1;

%e 1, 1365, 93093, 376805, 93093, 1365, 1; ...

%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=4: */ {a(n)=subst(B_q(n),q,4)}

%Y Cf. A022168, A125810, A125811, A125812, A125813, A125815.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 10 2006

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Last modified January 17 11:22 EST 2022. Contains 350393 sequences. (Running on oeis4.)