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A129276
Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.
3
1, 1, 1, 1, 2, 1, 1, 8, 8, 1, 1, 42, 106, 42, 1, 1, 241, 1558, 1558, 241, 1, 1, 1444, 23589, 53612, 23589, 1444, 1, 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1, 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1
OFFSET
0,5
COMMENTS
Dual triangle is A129274.
Central terms form a bisection of A127728.
LINKS
Eric Weisstein's World of Mathematics, q-Factorial.
FORMULA
T(n,k) = [q^(nk-k)] Product_{i=1..n} { (1-q^i)/(1-q) }^2 for n>0, with T(0,0)=1.
Row sums = (n!)^2/(n-1) for n>=2.
EXAMPLE
Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 4 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, (42), 71, 96, (106), 96, 71, (42), 19, 6, (1)],
where the terms enclosed in parenthesis form row 4.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 8, 8, 1;
1, 42, 106, 42, 1;
1, 241, 1558, 1558, 241, 1;
1, 1444, 23589, 53612, 23589, 1444, 1;
1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...
MATHEMATICA
faq[n_, q_] := Product[(1-q^k)/(1-q), {k, 1, n}]; t[0, 0] = t[1, 0] = t[1, 1] = 1; t[n_, k_] := SeriesCoefficient[faq[n, q]^2, {q, 0, (n-1)*k}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)
PROG
(PARI) T(n, k)=if(n==0, 1, polcoeff(prod(i=1, n, (1-x^i)/(1-x))^2, (n-1)*k))
CROSSREFS
Cf. A129277 (column 1), A129278 (column 2); A127728 (central terms), related triangles: A129274, A128564, A008302 (Mahonian numbers).
Sequence in context: A051428 A374147 A176698 * A156901 A167400 A225848
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 07 2007
STATUS
approved