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%I #17 Jun 12 2023 09:23:18
%S 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,
%T 119,119,47,7,1,1,8,72,256,390,256,72,8,1,1,9,104,500,1070,1070,500,
%U 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
%N Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows.
%C 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).
%H Paul D. Hanna, <a href="/A125806/b125806.txt">Rows n = 0..45, listed in flattened form as n, a(n), for n = 0..1080.</a>
%e The triangle of q-binomial coefficients:
%e C_q(n,k) = [Product_{i=n-k+1..n}(1-q^i)]/[Product_{j=1..k}(1-q^j)]
%e begins:
%e 1;
%e 1, 1;
%e 1, 1+q, 1;
%e 1, 1+q+q^2, 1+q+q^2, 1;
%e 1, 1+q+q^2+q^3, 1+q+2*q^2+q^3+q^4, 1+q+q^2+q^3, 1; ...
%e recurrence: C_q(n+1,k) = C_q(n,k-1) + q^k * C_q(n,k).
%e Element T(n,k) of this triangle equals the sum of the squares
%e of the coefficients of q in q-binomial coefficient C_q(n,k).
%e This triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 8, 4, 1;
%e 1, 5, 16, 16, 5, 1;
%e 1, 6, 29, 48, 29, 6, 1;
%e 1, 7, 47, 119, 119, 47, 7, 1;
%e 1, 8, 72, 256, 390, 256, 72, 8, 1;
%e 1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1;
%e 1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1;
%e 1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1;
%e 1, 12, 256, 2456, 11483, 28160, 37834, 28160, 11483, 2456, 256, 12, 1;
%e The central terms equal A063075:
%e 1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, ...
%e MATRIX INVERSE.
%e The matrix inverse starts
%e 1;
%e -1,1;
%e 1,-2,1;
%e -1,3,-3,1;
%e -1,0,4,-4,1;
%e 9,-21,12,4,-5,1;
%e -1,34,-73,44,1,-6,1;
%e -219,479,-219,-139,109,-5,-7,1;
%e 101,-1536,3072,-1776,-54,216,-16,-8,1; - _R. J. Mathar_, Mar 22 2013
%p C := proc(q,n,k)
%p local i,j ;
%p mul(1-q^i,i=n-k+1..n)/mul(1-q^j,j=1..k) ;
%p expand(factor(%)) ;
%p end proc:
%p A125806 := proc(n,k)
%p local qbin ,q;
%p qbin := [coeffs(C(q,n,k),q)] ;
%p add( e^2,e=qbin) ;
%p end proc: # _R. J. Mathar_, Mar 22 2013
%t T[n_, k_] := Module[{cc, q}, cc = CoefficientList[QBinomial[n, k, q] // FunctionExpand, q]; cc.cc];
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 08 2023 *)
%o (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)
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A063075 (central terms); A125807, A125808, A125809 (row sums).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Dec 11 2006
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