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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.

%I A125814 #26 Mar 15 2025 14:58:21
%S A125814 1,1,2,8,72,1552,84416,12107584,4726583424,5150624868864,
%T A125814 16010990175691264,144648776120641766400,3857411545088966609514496,
%U A125814 307705704204270334224705015808,74294186209325019487040708053442560,54874536782175258883045772243829235417088
%N A125814 q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4.
%H A125814 Alois P. Heinz, <a href="/A125814/b125814.txt">Table of n, a(n) for n = 0..65</a>
%F A125814 a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) for n>0, with a(0)=1.
%F A125814 a(n) = Sum_{k>=0} 4^k * A125810(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A125814 The recurrence: a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) is illustrated by:
%e A125814   a(2) = 1*(1) + 5*(1) + 1*(2) = 8;
%e A125814   a(3) = 1*(1) + 21*(1) + 21*(2) + 1*(8) = 72;
%e A125814   a(4) = 1*(1) + 85*(1) + 357*(2) + 85*(8) + 1*(72) = 1552.
%e A125814 Triangle A022168 begins:
%e A125814   1;
%e A125814   1,    1;
%e A125814   1,    5,     1;
%e A125814   1,   21,    21,      1;
%e A125814   1,   85,   357,     85,     1;
%e A125814   1,  341,  5797,   5797,   341,    1;
%e A125814   1, 1365, 93093, 376805, 93093, 1365, 1;
%e A125814   ...
%p A125814 b:= proc(o, u, t) option remember;
%p A125814      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(4^(u+j-1)*
%p A125814         b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
%p A125814     end:
%p A125814 a:= n-> b(n, 0$2):
%p A125814 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 21 2025
%t A125814 b[o_, u_, t_] := b[o, u, t] =
%t A125814    If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[4^(u + j - 1)*
%t A125814    b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
%t A125814 a[n_] := b[n, 0, 0];
%t A125814 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Mar 15 2025, after _Alois P. Heinz_ *)
%o A125814 (PARI)
%o A125814 /* q-Binomial coefficients: */
%o A125814 {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 A125814 /* q-Bell numbers = eigensequence of q-binomial triangle: */
%o A125814 {B_q(n)=if(n==0,1,sum(k=0,n-1,B_q(k)*C_q(n-1,k)))}
%o A125814 /* Eigensequence at q=4: */
%o A125814 {a(n)=subst(B_q(n),q,4)}
%Y A125814 Cf. A022168, A125810, A125811, A125812, A125813, A125815.
%Y A125814 Column k=4 of A381369.
%K A125814 nonn
%O A125814 0,3
%A A125814 _Paul D. Hanna_, Dec 10 2006