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A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

%I A125813 #22 Mar 15 2025 18:45:34
%S A125813 1,1,2,7,47,628,17327,1022983,132615812,38522717107,25526768401271,
%T A125813 39190247441314450,141213238745969102393,1207367655155905204747681,
%U A125813 24733467452839301566047854678,1224709126636123500201799360630423,147747406166666863538672620806542995763
%N A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.
%H A125813 Alois P. Heinz, <a href="/A125813/b125813.txt">Table of n, a(n) for n = 0..72</a>
%F A125813 a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) for n>0, with a(0)=1.
%F A125813 a(n) = Sum_{k>=0} 3^k * A125810(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A125813 The recurrence: a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) is illustrated by:
%e A125813   a(2) = 1*(1) + 4*(1) + 1*(2) = 7;
%e A125813   a(3) = 1*(1) + 13*(1) + 13*(2) + 1*(7) = 47;
%e A125813   a(4) = 1*(1) + 40*(1) + 130*(2) + 40*(7) + 1*(47) = 628.
%e A125813 Triangle A022167 begins:
%e A125813   1;
%e A125813   1, 1;
%e A125813   1, 4, 1;
%e A125813   1, 13, 13, 1;
%e A125813   1, 40, 130, 40, 1;
%e A125813   1, 121, 1210, 1210, 121, 1;
%e A125813   1, 364, 11011, 33880, 11011, 364, 1;
%e A125813   ...
%p A125813 b:= proc(o, u, t) option remember;
%p A125813      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(3^(u+j-1)*
%p A125813         b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
%p A125813     end:
%p A125813 a:= n-> b(n, 0$2):
%p A125813 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 21 2025
%t A125813 b[o_, u_, t_] := b[o, u, t] =
%t A125813    If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[3^(u + j - 1)*
%t A125813    b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
%t A125813 a[n_] := b[n, 0, 0];
%t A125813 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Mar 15 2025, after _Alois P. Heinz_ *)
%o A125813 (PARI) /* q-Binomial coefficients: */
%o A125813 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 A125813 /* q-Bell numbers = eigensequence of q-binomial triangle: */
%o A125813 B_q(n)=if(n==0,1,sum(k=0,n-1,B_q(k)*C_q(n-1,k)))
%o A125813 /* Eigensequence at q=3: */
%o A125813 a(n)=subst(B_q(n),q,3)
%Y A125813 Cf. A022167, A125810, A125811, A125812, A125814, A125815.
%Y A125813 Column k=3 of A381369.
%K A125813 nonn
%O A125813 0,3
%A A125813 _Paul D. Hanna_, Dec 10 2006