A125806 Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 29, 48, 29, 6, 1, 1, 7, 47, 119, 119, 47, 7, 1, 1, 8, 72, 256, 390, 256, 72, 8, 1, 1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1, 1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1, 1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1

Offset

0,5

Comments

Central terms equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

Links

Paul D. Hanna, Rows n = 0..45, listed in flattened form as n, a(n), for n = 0..1080.

Example

The triangle of q-binomial coefficients:
C_q(n,k) = [Product_{i=n-k+1..n}(1-q^i)]/[Product_{j=1..k}(1-q^j)]
begins:
1;
1, 1;
1, 1+q, 1;
1, 1+q+q^2, 1+q+q^2, 1;
1, 1+q+q^2+q^3, 1+q+2*q^2+q^3+q^4, 1+q+q^2+q^3, 1; ...
recurrence: C_q(n+1,k) = C_q(n,k-1) + q^k * C_q(n,k).
Element T(n,k) of this triangle equals the sum of the squares
of the coefficients of q in q-binomial coefficient C_q(n,k).
This triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 8, 4, 1;
1, 5, 16, 16, 5, 1;
1, 6, 29, 48, 29, 6, 1;
1, 7, 47, 119, 119, 47, 7, 1;
1, 8, 72, 256, 390, 256, 72, 8, 1;
1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1;
1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1;
1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1;
1, 12, 256, 2456, 11483, 28160, 37834, 28160, 11483, 2456, 256, 12, 1;
The central terms equal A063075:
1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
1,-2,1;
-1,3,-3,1;
-1,0,4,-4,1;
9,-21,12,4,-5,1;
-1,34,-73,44,1,-6,1;
-219,479,-219,-139,109,-5,-7,1;
101,-1536,3072,-1776,-54,216,-16,-8,1; - R. J. Mathar, Mar 22 2013

Maple

C := proc(q, n, k)
    local i, j ;
    mul(1-q^i, i=n-k+1..n)/mul(1-q^j, j=1..k) ;
    expand(factor(%)) ;
end proc:
A125806 := proc(n, k)
    local qbin , q;
    qbin := [coeffs(C(q, n, k), q)] ;
    add( e^2, e=qbin) ;
end proc: # R. J. Mathar, Mar 22 2013

Mathematica

T[n_, k_] := Module[{cc, q}, cc = CoefficientList[QBinomial[n, k, q] // FunctionExpand, q]; cc.cc];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 08 2023 *)

Prog

(PARI) T(n, k)=local(C_q=if(n==0 || k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j))); sum(i=0, (n-k)*k, polcoeff(C_q, i)^2)
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A063075 (central terms); A125807, A125808, A125809 (row sums).

Sequence in context: A300260 A026692 A114202 * A347148 A202756 A156354

Adjacent sequences: A125803 A125804 A125805 * A125807 A125808 A125809

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 11 2006

Status

approved