A125815 - OEIS

%I #6 Jun 14 2017 00:19:28

%S 1,1,2,9,103,3276,307867,89520089,83657942588,258923776689771,

%T 2717711483011792407,98702105953049319472394,

%U 12629828399521800714941435773,5784963467206342855747483263957541

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

%F a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k) for n>0, with a(0)=1.

%e The recurrence: a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k)

%e is illustrated by:

%e a(2) = 1*(1) + 6*(1) + 1*(2) = 9;

%e a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;

%e a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.

%e Triangle A022169 begins:

%e 1;

%e 1, 1;

%e 1, 6, 1;

%e 1, 31, 31, 1;

%e 1, 156, 806, 156, 1;

%e 1, 781, 20306, 20306, 781, 1;

%e 1, 3906, 508431, 2558556, 508431, 3906, 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=5: */ {a(n)=subst(B_q(n),q,5)}

%Y Cf. A022169, A125810, A125811, A125812, A125813, A125814.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 10 2006