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 A125806 Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows. 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 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). 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 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, ... 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 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 * A202756 A156354 A295205 Adjacent sequences:  A125803 A125804 A125805 * A125807 A125808 A125809 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 11 2006 STATUS approved

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Last modified April 25 13:31 EDT 2019. Contains 322461 sequences. (Running on oeis4.)