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
Examples
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
Links
Programs
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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 *) -
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(""))
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