A128080 - OEIS

%I #29 Sep 27 2022 08:45:35

%S 1,1,1,1,1,1,2,3,3,3,2,1,1,3,6,9,12,14,15,14,12,9,6,3,1,1,4,10,19,31,

%T 45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1,1,5,15,34,65,110,170,

%U 244,330,424,521,614,696,760,801,815,801,760,696,614,521,424,330,244,170

%N Triangle, read by rows of n(n-1)+1 terms, of coefficients of q in the q-analog of the odd double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

%C See A128084 for the triangle of coefficients of q in the q-analog of the even double factorials.

%H Paul D. Hanna, <a href="/A128080/b128080.txt">Rows n=0..31 of triangle, in flattened form.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.

%F The row sums are A001147, the odd double factorial numbers (2n-1)!!.

%e Triangle begins:

%e   1;

%e   1;

%e   1,1,1;

%e   1,2,3,3,3,2,1;

%e   1,3,6,9,12,14,15,14,12,9,6,3,1;

%e   1,4,10,19,31,45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;

%e   ...

%p b:= proc(n) option remember; `if`(n=0, 1,

%p       simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(b(n)):

%p seq(T(n), n=0..6);  # _Alois P. Heinz_, Sep 22 2021

%t Catenate@Table[CoefficientList[Cancel@FunctionExpand[-q QPochhammer[1/q, q^2, n + 1]/(1 - q)^(n + 1)], q], {n, 0, 6}] (* _Vladimir Reshetnikov_, Sep 22 2021 *)

%t T[n_] := If[n == 0, {1}, Product[(1 - q^(2 j - 1))/(1 - q), {j, 1, n}] + O[q]^(n (n + 1)) // CoefficientList[#, q]&];

%t Table[T[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Sep 27 2022 *)

%o (PARI) T(n,k)=if(k<0 || k>n*(n-1),0,if(n==0,1,polcoeff(prod(j=1,n,(1-q^(2*j-1))/(1-q)),k,q)))

%o for(n=0,8,for(k=0,n*(n-1),print1(T(n,k),", "));print(""))

%Y Cf. A001147 (row sums); A128081 (central terms), A128082 (diagonal), A128083 (row squared sums); A128084.

%K nonn,tabf

%O 0,7

%A _Paul D. Hanna_, Feb 14 2007