A128080 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.
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, 45, 60, 74, 86, 94, 97, 94, 86, 74, 60, 45, 31, 19, 10, 4, 1, 1, 5, 15, 34, 65, 110, 170, 244, 330, 424, 521, 614, 696, 760, 801, 815, 801, 760, 696, 614, 521, 424, 330, 244, 170
Offset: 0
Examples
Triangle begins: 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; 1,4,10,19,31,45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1; ...
Links
- Paul D. Hanna, Rows n=0..31 of triangle, in flattened form.
- Eric Weisstein's World of Mathematics, q-Factorial.
Crossrefs
Programs
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Maple
b:= proc(n) option remember; `if`(n=0, 1, simplify(b(n-1)*(1-q^(2*n-1))/(1-q))) end: T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(b(n)): seq(T(n), n=0..6); # Alois P. Heinz, Sep 22 2021 -
Mathematica
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[n_] := If[n == 0, {1}, Product[(1 - q^(2 j - 1))/(1 - q), {j, 1, n}] + O[q]^(n (n + 1)) // CoefficientList[#, q]&]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Sep 27 2022 *) -
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))) for(n=0,8,for(k=0,n*(n-1),print1(T(n,k),", "));print(""))
Formula
The row sums are A001147, the odd double factorial numbers (2n-1)!!.
Comments