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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. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
See A128084 for the triangle of coefficients of q in the q-analog of the even double factorials.
LINKS
Eric Weisstein's World of Mathematics, q-Factorial.
FORMULA
The row sums are A001147, the odd double factorial numbers (2n-1)!!.
EXAMPLE
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;
...
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 *)
PROG
(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(""))
CROSSREFS
Cf. A001147 (row sums); A128081 (central terms), A128082 (diagonal), A128083 (row squared sums); A128084.
Sequence in context: A143605 A098418 A364367 * A062187 A334485 A347824
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Feb 14 2007
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)