A193762 - OEIS

%I #17 Aug 10 2020 13:30:11

%S 1,2,1,5,7,4,1,14,36,45,35,18,6,1,42,165,330,440,433,330,198,93,33,8,

%T 1,132,715,2002,3822,5551,6496,6331,5239,3718,2268,1183,520,187,52,10,

%U 1,429,3003,10920,27300,52500,82476,109800,126885,129360,117635,96096,70800,47100,28245,15195,7269,3048,1095,325,75,12,1,1430,12376,55692

%N Triangle read by rows giving coefficients of Genocchi q-numbers B_n(1,q) (n >= 1) expanded in powers of q.

%C Row lengths are 1, 2, 4, 7, 11, 16, 22, 29, ...

%H Han, Guo-Niu; and Zeng, Jiang; <a href="https://doi.org/10.1016/S0012-365X(97)00189-1">q-polynômes de Gandhi et statistique de Denert</a>, Discrete Math. 205 (1999), 119-143. See p. 140.

%e    1

%e    2 +    q

%e    5 +  7*q +  4*q^2 +    q^3

%e   14 + 36*q + 45*q^2 + 35*q^3 + 18*q^4 + 6*q^5 + q^6

%e   ...

%p qBrack := proc(n,q)

%p         add(q^i,i=0..n-1) ;

%p end proc:

%p B := proc(n,x,q)

%p         a :=0 ;

%p         for npr from 1 to n do

%p                 q^(npr-1)*(1+q*x-x)*t^npr*mul((qBrack(k,q)+q^k*x)^2,k=1..npr-1) ;

%p                 %/mul(q^(k-1)*(1+q*x-x)+t*(qBrack(k-1,q)+q^(k-1)*x)^2,k=1..npr) ;

%p                 a := a+coeftayl(%,t=0,n) ;

%p         end do;

%p         expand(a) ;

%p end proc:

%p A193762_row := proc(n)

%p         B(n,1,q) ;

%p end proc:

%p A193762 := proc(n,k)

%p         coeftayl( A193762_row(n),q=0,k) ;

%p end proc: # _R. J. Mathar_, Oct 02 2011

%K nonn,tabf

%O 1,2

%A _N. J. A. Sloane_, Aug 04 2011