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A125815 q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5.

%I A125815 #28 Aug 07 2025 17:16:59
%S A125815 1,1,2,9,103,3276,307867,89520089,83657942588,258923776689771,
%T A125815 2717711483011792407,98702105953049319472394,
%U A125815 12629828399521800714941435773,5784963467206342855747483263957541,9613516698678314330032600987632336641122,58637855728567773833514895771659795097103477549
%N A125815 q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5.
%H A125815 Alois P. Heinz, <a href="/A125815/b125815.txt">Table of n, a(n) for n = 0..60</a>
%F A125815 a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k) for n>0, with a(0)=1.
%F A125815 a(n) = Sum_{k>=0} 5^k * A125810(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A125815 The recurrence: a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k)
%e A125815 is illustrated by:
%e A125815 a(2) = 1*(1) + 6*(1) + 1*(2) = 9;
%e A125815 a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;
%e A125815 a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.
%e A125815 Triangle A022169 begins:
%e A125815   1;
%e A125815   1,    1;
%e A125815   1,    6,      1;
%e A125815   1,   31,     31,       1;
%e A125815   1,  156,    806,     156,      1;
%e A125815   1,  781,  20306,   20306,    781,    1;
%e A125815   1, 3906, 508431, 2558556, 508431, 3906, 1;
%e A125815   ...
%p A125815 b:= proc(o, u, t) option remember;
%p A125815      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(5^(u+j-1)*
%p A125815         b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
%p A125815     end:
%p A125815 a:= n-> b(n, 0$2):
%p A125815 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 21 2025
%t A125815 b[o_, u_, t_] := b[o, u, t] =
%t A125815    If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[5^(u + j - 1)*
%t A125815    b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
%t A125815 a[n_] := b[n, 0, 0];
%t A125815 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Mar 15 2025, after _Alois P. Heinz_ *)
%o A125815 (PARI)
%o A125815 /* q-Binomial coefficients: */
%o A125815 {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 A125815 /* q-Bell numbers = eigensequence of q-binomial triangle: */
%o A125815 {B_q(n)=if(n==0,1,sum(k=0,n-1,B_q(k)*C_q(n-1,k)))}
%o A125815 /* Eigensequence at q=5: */
%o A125815 {a(n)=subst(B_q(n),q,5)}
%Y A125815 Cf. A022169, A125810, A125811, A125812, A125813, A125814.
%Y A125815 Column k=5 of A381369.
%K A125815 nonn
%O A125815 0,3
%A A125815 _Paul D. Hanna_, Dec 10 2006